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Theorem locfinreflem 29235
 Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
Hypotheses
Ref Expression
locfinref.x 𝑋 = 𝐽
locfinref.1 (𝜑𝑈𝐽)
locfinref.2 (𝜑𝑋 = 𝑈)
locfinref.3 (𝜑𝑉𝐽)
locfinref.4 (𝜑𝑉Ref𝑈)
locfinref.5 (𝜑𝑉 ∈ (LocFin‘𝐽))
Assertion
Ref Expression
locfinreflem (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Distinct variable groups:   𝑓,𝐽   𝑈,𝑓   𝑓,𝑉   𝜑,𝑓
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem locfinreflem
Dummy variables 𝑔 𝑗 𝑘 𝑢 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfinref.4 . . . 4 (𝜑𝑉Ref𝑈)
2 locfinref.5 . . . . 5 (𝜑𝑉 ∈ (LocFin‘𝐽))
3 reff 29234 . . . . 5 (𝑉 ∈ (LocFin‘𝐽) → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
42, 3syl 17 . . . 4 (𝜑 → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
51, 4mpbid 221 . . 3 (𝜑 → ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))))
65simprd 478 . 2 (𝜑 → ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))
7 funmpt 5840 . . . . . 6 Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
87a1i 11 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
9 eqid 2610 . . . . . . 7 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
109dmmptss 5548 . . . . . 6 dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ ran 𝑔
11 frn 5966 . . . . . . 7 (𝑔:𝑉𝑈 → ran 𝑔𝑈)
1211ad2antlr 759 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran 𝑔𝑈)
1310, 12syl5ss 3579 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈)
14 locfintop 21134 . . . . . . . . . 10 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
152, 14syl 17 . . . . . . . . 9 (𝜑𝐽 ∈ Top)
1615ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝐽 ∈ Top)
17 cnvimass 5404 . . . . . . . . . 10 (𝑔 “ {𝑢}) ⊆ dom 𝑔
18 fdm 5964 . . . . . . . . . . 11 (𝑔:𝑉𝑈 → dom 𝑔 = 𝑉)
1918ad3antlr 763 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → dom 𝑔 = 𝑉)
2017, 19syl5sseq 3616 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝑉)
21 locfinref.3 . . . . . . . . . 10 (𝜑𝑉𝐽)
2221ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝑉𝐽)
2320, 22sstrd 3578 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝐽)
24 uniopn 20527 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔 “ {𝑢}) ⊆ 𝐽) → (𝑔 “ {𝑢}) ∈ 𝐽)
2516, 23, 24syl2anc 691 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ∈ 𝐽)
2625ralrimiva 2949 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽)
279rnmptss 6299 . . . . . 6 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
2826, 27syl 17 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
29 eqid 2610 . . . . . . . . . 10 𝑉 = 𝑉
30 eqid 2610 . . . . . . . . . 10 𝑈 = 𝑈
3129, 30refbas 21123 . . . . . . . . 9 (𝑉Ref𝑈 𝑈 = 𝑉)
321, 31syl 17 . . . . . . . 8 (𝜑 𝑈 = 𝑉)
3332ad2antrr 758 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = 𝑉)
34 nfv 1830 . . . . . . . . . . . . 13 𝑣(𝜑𝑔:𝑉𝑈)
35 nfra1 2925 . . . . . . . . . . . . 13 𝑣𝑣𝑉 𝑣 ⊆ (𝑔𝑣)
3634, 35nfan 1816 . . . . . . . . . . . 12 𝑣((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
37 nfre1 2988 . . . . . . . . . . . 12 𝑣𝑣𝑉 𝑥𝑣
3836, 37nfan 1816 . . . . . . . . . . 11 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣)
39 ffn 5958 . . . . . . . . . . . . . . 15 (𝑔:𝑉𝑈𝑔 Fn 𝑉)
4039ad4antlr 765 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
41 simplr 788 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑣𝑉)
42 fnfvelrn 6264 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑉𝑣𝑉) → (𝑔𝑣) ∈ ran 𝑔)
4340, 41, 42syl2anc 691 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → (𝑔𝑣) ∈ ran 𝑔)
44 ssid 3587 . . . . . . . . . . . . . . 15 𝑣𝑣
4539ad3antlr 763 . . . . . . . . . . . . . . . 16 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑔 Fn 𝑉)
46 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑔𝑣) = (𝑔𝑣)
47 fniniseg 6246 . . . . . . . . . . . . . . . . . 18 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {(𝑔𝑣)}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))))
4847biimpar 501 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑉 ∧ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
4946, 48mpanr2 716 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑉𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
5045, 49sylancom 698 . . . . . . . . . . . . . . 15 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
51 ssuni 4395 . . . . . . . . . . . . . . 15 ((𝑣𝑣𝑣 ∈ (𝑔 “ {(𝑔𝑣)})) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5244, 50, 51sylancr 694 . . . . . . . . . . . . . 14 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5352sselda 3568 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑥 (𝑔 “ {(𝑔𝑣)}))
54 sneq 4135 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝑔𝑣) → {𝑢} = {(𝑔𝑣)})
5554imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5655unieqd 4382 . . . . . . . . . . . . . . 15 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5756eleq2d 2673 . . . . . . . . . . . . . 14 (𝑢 = (𝑔𝑣) → (𝑥 (𝑔 “ {𝑢}) ↔ 𝑥 (𝑔 “ {(𝑔𝑣)})))
5857rspcev 3282 . . . . . . . . . . . . 13 (((𝑔𝑣) ∈ ran 𝑔𝑥 (𝑔 “ {(𝑔𝑣)})) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
5943, 53, 58syl2anc 691 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
6059adantllr 751 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
61 simpr 476 . . . . . . . . . . 11 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑣𝑉 𝑥𝑣)
6238, 60, 61r19.29af 3058 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
63 nfv 1830 . . . . . . . . . . . . . 14 𝑣 𝑢 ∈ ran 𝑔
6436, 63nfan 1816 . . . . . . . . . . . . 13 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔)
65 nfv 1830 . . . . . . . . . . . . 13 𝑣 𝑥 (𝑔 “ {𝑢})
6664, 65nfan 1816 . . . . . . . . . . . 12 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢}))
6720ad3antrrr 762 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → (𝑔 “ {𝑢}) ⊆ 𝑉)
68 simplr 788 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣 ∈ (𝑔 “ {𝑢}))
6967, 68sseldd 3569 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣𝑉)
70 simpr 476 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑥𝑣)
71 simpr 476 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → 𝑥 (𝑔 “ {𝑢}))
72 eluni2 4376 . . . . . . . . . . . . 13 (𝑥 (𝑔 “ {𝑢}) ↔ ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7371, 72sylib 207 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7466, 69, 70, 73reximd2a 2996 . . . . . . . . . . 11 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7574r19.29an 3059 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7662, 75impbida 873 . . . . . . . . 9 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (∃𝑣𝑉 𝑥𝑣 ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})))
77 eluni2 4376 . . . . . . . . 9 (𝑥 𝑉 ↔ ∃𝑣𝑉 𝑥𝑣)
78 eliun 4460 . . . . . . . . 9 (𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
7976, 77, 783bitr4g 302 . . . . . . . 8 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑥 𝑉𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8079eqrdv 2608 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑉 = 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
81 dfiun3g 5299 . . . . . . . 8 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8226, 81syl 17 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8333, 80, 823eqtrd 2648 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8411ad3antlr 763 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ran 𝑔𝑈)
85 vex 3176 . . . . . . . . . . 11 𝑤 ∈ V
869elrnmpt 5293 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8785, 86mp1i 13 . . . . . . . . . 10 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8887biimpa 500 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}))
89 ssrexv 3630 . . . . . . . . 9 (ran 𝑔𝑈 → (∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢})))
9084, 88, 89sylc 63 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}))
91 nfv 1830 . . . . . . . . . 10 𝑢((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
92 nfmpt1 4675 . . . . . . . . . . . 12 𝑢(𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9392nfrn 5289 . . . . . . . . . . 11 𝑢ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9493nfcri 2745 . . . . . . . . . 10 𝑢 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9591, 94nfan 1816 . . . . . . . . 9 𝑢(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
96 simpr 476 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤 = (𝑔 “ {𝑢}))
97 nfv 1830 . . . . . . . . . . . . . . . 16 𝑣 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9836, 97nfan 1816 . . . . . . . . . . . . . . 15 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
99 nfv 1830 . . . . . . . . . . . . . . 15 𝑣 𝑢𝑈
10098, 99nfan 1816 . . . . . . . . . . . . . 14 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈)
101 nfv 1830 . . . . . . . . . . . . . 14 𝑣 𝑤 = (𝑔 “ {𝑢})
102100, 101nfan 1816 . . . . . . . . . . . . 13 𝑣(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢}))
103 simp-5r 805 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
10439ad5antlr 767 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑔 Fn 𝑉)
105 fniniseg 6246 . . . . . . . . . . . . . . . . . . 19 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
106104, 105syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
107106biimpa 500 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢))
108107simpld 474 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑉)
109 rspa 2914 . . . . . . . . . . . . . . . 16 ((∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣) ∧ 𝑣𝑉) → 𝑣 ⊆ (𝑔𝑣))
110103, 108, 109syl2anc 691 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣 ⊆ (𝑔𝑣))
111107simprd 478 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑔𝑣) = 𝑢)
112110, 111sseqtrd 3604 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑢)
113112ex 449 . . . . . . . . . . . . 13 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) → 𝑣𝑢))
114102, 113ralrimi 2940 . . . . . . . . . . . 12 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
115 unissb 4405 . . . . . . . . . . . 12 ( (𝑔 “ {𝑢}) ⊆ 𝑢 ↔ ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
116114, 115sylibr 223 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑔 “ {𝑢}) ⊆ 𝑢)
11796, 116eqsstrd 3602 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤𝑢)
118117exp31 628 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (𝑢𝑈 → (𝑤 = (𝑔 “ {𝑢}) → 𝑤𝑢)))
11995, 118reximdai 2995 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤𝑢))
12090, 119mpd 15 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤𝑢)
121120ralrimiva 2949 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)
122 vex 3176 . . . . . . . . . 10 𝑔 ∈ V
123122rnex 6992 . . . . . . . . 9 ran 𝑔 ∈ V
124123mptex 6390 . . . . . . . 8 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V
125 rnexg 6990 . . . . . . . 8 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
126124, 125mp1i 13 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
127 eqid 2610 . . . . . . . 8 ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
128127, 30isref 21122 . . . . . . 7 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
129126, 128syl 17 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
13083, 121, 129mpbir2and 959 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈)
13115ad2antrr 758 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝐽 ∈ Top)
132 locfinref.2 . . . . . . . 8 (𝜑𝑋 = 𝑈)
133132ad2antrr 758 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = 𝑈)
134133, 83eqtrd 2644 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
135 nfv 1830 . . . . . . . . 9 𝑣 𝑥𝑋
13636, 135nfan 1816 . . . . . . . 8 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋)
137 simplr 788 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → 𝑣𝐽)
138 ffun 5961 . . . . . . . . . . . . . 14 (𝑔:𝑉𝑈 → Fun 𝑔)
139138ad6antlr 769 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → Fun 𝑔)
140 imafi 8142 . . . . . . . . . . . . 13 ((Fun 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
141139, 140sylancom 698 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
142 simp3 1056 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘))
143 sneq 4135 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑘 → {𝑢} = {𝑘})
144143imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑘 → (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
145144unieqd 4382 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑘 (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
146122cnvex 7006 . . . . . . . . . . . . . . . . . . . . . . 23 𝑔 ∈ V
147 imaexg 6995 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 ∈ V → (𝑔 “ {𝑘}) ∈ V)
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 “ {𝑘}) ∈ V
149148uniex 6851 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 “ {𝑘}) ∈ V
150145, 9, 149fvmpt 6191 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ran 𝑔 → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
1511503ad2ant2 1076 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
152142, 151eqtrd 2644 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = (𝑔 “ {𝑘}))
153152ineq1d 3775 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → (𝑤𝑣) = ( (𝑔 “ {𝑘}) ∩ 𝑣))
154153neeq1d 2841 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑤𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅))
155123a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → ran 𝑔 ∈ V)
156 imaexg 6995 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 ∈ V → (𝑔 “ {𝑢}) ∈ V)
157146, 156ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑔 “ {𝑢}) ∈ V
158157uniex 6851 . . . . . . . . . . . . . . . . . . 19 (𝑔 “ {𝑢}) ∈ V
159158, 9fnmpti 5935 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔
160 dffn4 6034 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔 ↔ (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
161159, 160mpbi 219 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
162161a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
163154, 155, 162rabfodom 28728 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅})
164 sneq 4135 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑢 → {𝑘} = {𝑢})
165164imaeq2d 5385 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
166165unieqd 4382 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
167166ineq1d 3775 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ( (𝑔 “ {𝑘}) ∩ 𝑣) = ( (𝑔 “ {𝑢}) ∩ 𝑣))
168167neeq1d 2841 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅))
169168cbvrabv 3172 . . . . . . . . . . . . . . 15 {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅} = {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅}
170163, 169syl6breq 4624 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅})
171123rabex 4740 . . . . . . . . . . . . . . 15 {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V
172 nfv 1830 . . . . . . . . . . . . . . . . . . . . 21 𝑗(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣)
173 nfrab1 3099 . . . . . . . . . . . . . . . . . . . . . 22 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}
174173nfel1 2765 . . . . . . . . . . . . . . . . . . . . 21 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin
175172, 174nfan 1816 . . . . . . . . . . . . . . . . . . . 20 𝑗((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)
176 nfv 1830 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑢 ∈ ran 𝑔
177175, 176nfan 1816 . . . . . . . . . . . . . . . . . . 19 𝑗(((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔)
178 nfv 1830 . . . . . . . . . . . . . . . . . . 19 𝑗( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅
179177, 178nfan 1816 . . . . . . . . . . . . . . . . . 18 𝑗((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅)
180 nfv 1830 . . . . . . . . . . . . . . . . . . 19 𝑗(𝑔𝑘) = 𝑢
181173, 180nfrex 2990 . . . . . . . . . . . . . . . . . 18 𝑗𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢
18239ad5antlr 767 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
183182ad5antr 766 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑔 Fn 𝑉)
184 simplr 788 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ (𝑔 “ {𝑢}))
185 fniniseg 6246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 Fn 𝑉 → (𝑗 ∈ (𝑔 “ {𝑢}) ↔ (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢)))
186185biimpa 500 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 Fn 𝑉𝑗 ∈ (𝑔 “ {𝑢})) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
187183, 184, 186syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
188187simpld 474 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗𝑉)
189 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑣) ≠ ∅)
190 rabid 3095 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ↔ (𝑗𝑉 ∧ (𝑗𝑣) ≠ ∅))
191188, 189, 190sylanbrc 695 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})
192187simprd 478 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑔𝑗) = 𝑢)
193 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
194193eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝑔𝑘) = 𝑢 ↔ (𝑔𝑗) = 𝑢))
195194rspcev 3282 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∧ (𝑔𝑗) = 𝑢) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
196191, 192, 195syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
197 uniinn0 28749 . . . . . . . . . . . . . . . . . . . 20 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ ↔ ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
198197biimpi 205 . . . . . . . . . . . . . . . . . . 19 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
199198adantl 481 . . . . . . . . . . . . . . . . . 18 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
200179, 181, 196, 199r19.29af2 3057 . . . . . . . . . . . . . . . . 17 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
201200ex 449 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) → (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢))
202201ss2rabdv 3646 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
203 ssdomg 7887 . . . . . . . . . . . . . . 15 ({𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V → ({𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}))
204171, 202, 203mpsyl 66 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
205 domtr 7895 . . . . . . . . . . . . . 14 (({𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ∧ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
206170, 204, 205syl2anc 691 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
207182adantr 480 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → 𝑔 Fn 𝑉)
208 dffn3 5967 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
209208biimpi 205 . . . . . . . . . . . . . 14 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
210 ssrab2 3650 . . . . . . . . . . . . . . 15 {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉
211 fimarab 28825 . . . . . . . . . . . . . . 15 ((𝑔:𝑉⟶ran 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
212210, 211mpan2 703 . . . . . . . . . . . . . 14 (𝑔:𝑉⟶ran 𝑔 → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
213207, 209, 2123syl 18 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
214206, 213breqtrrd 4611 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}))
215 domfi 8066 . . . . . . . . . . . 12 (((𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
216141, 214, 215syl2anc 691 . . . . . . . . . . 11 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
217216ex 449 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → ({𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
218217imdistanda 725 . . . . . . . . 9 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) → ((𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
219218imp 444 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
220 simplll 794 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → 𝜑)
221 locfinref.x . . . . . . . . . . . . 13 𝑋 = 𝐽
222221, 29islocfin 21130 . . . . . . . . . . . 12 (𝑉 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
2232, 222sylib 207 . . . . . . . . . . 11 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
224223simp3d 1068 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
225224r19.21bi 2916 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
226220, 225sylancom 698 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
227136, 137, 219, 226reximd2a 2996 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
228227ralrimiva 2949 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
229221, 127islocfin 21130 . . . . . 6 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
230131, 134, 228, 229syl3anbrc 1239 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))
231 funeq 5823 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (Fun 𝑓 ↔ Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))))
232 dmeq 5246 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → dom 𝑓 = dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
233232sseq1d 3595 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (dom 𝑓𝑈 ↔ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈))
234 rneq 5272 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ran 𝑓 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
235234sseq1d 3595 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓𝐽 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽))
236231, 233, 2353anbi123d 1391 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ↔ (Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)))
237234breq1d 4593 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓Ref𝑈 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈))
238234eleq1d 2672 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))
239237, 238anbi12d 743 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))))
240236, 239anbi12d 743 . . . . . 6 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) ↔ ((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))))
241124, 240spcev 3273 . . . . 5 (((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
2428, 13, 28, 130, 230, 241syl32anc 1326 . . . 4 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
243242expl 646 . . 3 (𝜑 → ((𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
244243exlimdv 1848 . 2 (𝜑 → (∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
2456, 244mpd 15 1 (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804   ≼ cdom 7839  Fincfn 7841  Topctop 20517  Refcref 21115  LocFinclocfin 21117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822  df-top 20521  df-ref 21118  df-locfin 21120 This theorem is referenced by:  locfinref  29236
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