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Theorem rnelfmlem 21566
 Description: Lemma for rnelfm 21567. (Contributed by Jeff Hankins, 14-Nov-2009.)
Assertion
Ref Expression
rnelfmlem (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐿   𝑥,𝑋   𝑥,𝑌

Proof of Theorem rnelfmlem
Dummy variables 𝑟 𝑠 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1059 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
2 cnvimass 5404 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
3 fdm 5964 . . . . . . . 8 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
42, 3syl5sseq 3616 . . . . . . 7 (𝐹:𝑌𝑋 → (𝐹𝑥) ⊆ 𝑌)
51, 4syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ⊆ 𝑌)
6 simpl1 1057 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
7 elpw2g 4754 . . . . . . 7 (𝑌𝐴 → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
86, 7syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
95, 8mpbird 246 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
109adantr 480 . . . 4 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
11 eqid 2610 . . . 4 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
1210, 11fmptd 6292 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌)
13 frn 5966 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌 → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
1412, 13syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
15 filtop 21469 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
16153ad2ant2 1076 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
1716adantr 480 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
18 fimacnv 6255 . . . . . . . . 9 (𝐹:𝑌𝑋 → (𝐹𝑋) = 𝑌)
1918eqcomd 2616 . . . . . . . 8 (𝐹:𝑌𝑋𝑌 = (𝐹𝑋))
20193ad2ant3 1077 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌 = (𝐹𝑋))
2120adantr 480 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 = (𝐹𝑋))
22 imaeq2 5381 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322eqeq2d 2620 . . . . . . 7 (𝑥 = 𝑋 → (𝑌 = (𝐹𝑥) ↔ 𝑌 = (𝐹𝑋)))
2423rspcev 3282 . . . . . 6 ((𝑋𝐿𝑌 = (𝐹𝑋)) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2517, 21, 24syl2anc 691 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2611elrnmpt 5293 . . . . . . 7 (𝑌𝐴 → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
27263ad2ant1 1075 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2827adantr 480 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2925, 28mpbird 246 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
30 ne0i 3880 . . . 4 (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
3129, 30syl 17 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
32 0nelfil 21463 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐿)
33323ad2ant2 1076 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → ¬ ∅ ∈ 𝐿)
3433adantr 480 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ 𝐿)
35 0ex 4718 . . . . . . 7 ∅ ∈ V
3611elrnmpt 5293 . . . . . . 7 (∅ ∈ V → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥)))
3735, 36ax-mp 5 . . . . . 6 (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥))
38 ffn 5958 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
39 fvelrnb 6153 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
41403ad2ant3 1077 . . . . . . . . . . . . . . . 16 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4241ad2antrr 758 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
43 eleq1 2676 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
4443biimparc 503 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥 ∧ (𝐹𝑧) = 𝑦) → (𝐹𝑧) ∈ 𝑥)
4544ad2ant2l 778 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝐿𝑦𝑥) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
4645adantll 746 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
47 ffun 5961 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:𝑌𝑋 → Fun 𝐹)
48473ad2ant3 1077 . . . . . . . . . . . . . . . . . . . 20 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
4948ad3antrrr 762 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → Fun 𝐹)
503eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝑌𝑋 → (𝑧 ∈ dom 𝐹𝑧𝑌))
5150biimpar 501 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝑌𝑋𝑧𝑌) → 𝑧 ∈ dom 𝐹)
52513ad2antl3 1218 . . . . . . . . . . . . . . . . . . . . 21 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5352adantlr 747 . . . . . . . . . . . . . . . . . . . 20 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5453ad2ant2r 779 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ dom 𝐹)
55 fvimacnv 6240 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5649, 54, 55syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5746, 56mpbid 221 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹𝑥))
58 n0i 3879 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐹𝑥) → ¬ (𝐹𝑥) = ∅)
59 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = ∅ ↔ ∅ = (𝐹𝑥))
6058, 59sylnib 317 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐹𝑥) → ¬ ∅ = (𝐹𝑥))
6157, 60syl 17 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ¬ ∅ = (𝐹𝑥))
6261rexlimdvaa 3014 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → ¬ ∅ = (𝐹𝑥)))
6342, 62sylbid 229 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 → ¬ ∅ = (𝐹𝑥)))
6463con2d 128 . . . . . . . . . . . . 13 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹))
6564expr 641 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑦𝑥 → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹)))
6665com23 84 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (∅ = (𝐹𝑥) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹)))
6766impr 647 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
6867alrimiv 1842 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
69 imnan 437 . . . . . . . . . . . 12 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ (𝑦𝑥𝑦 ∈ ran 𝐹))
70 elin 3758 . . . . . . . . . . . 12 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
7169, 70xchbinxr 324 . . . . . . . . . . 11 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
7271albii 1737 . . . . . . . . . 10 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
73 eq0 3888 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
74 eqcom 2617 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∅ = (𝑥 ∩ ran 𝐹))
7572, 73, 743bitr2i 287 . . . . . . . . 9 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∅ = (𝑥 ∩ ran 𝐹))
7668, 75sylib 207 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ = (𝑥 ∩ ran 𝐹))
77 simpll2 1094 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝐿 ∈ (Fil‘𝑋))
78 simprl 790 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝑥𝐿)
79 simplr 788 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ran 𝐹𝐿)
80 filin 21468 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8177, 78, 79, 80syl3anc 1318 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8276, 81eqeltrd 2688 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ ∈ 𝐿)
8382rexlimdvaa 3014 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 ∅ = (𝐹𝑥) → ∅ ∈ 𝐿))
8437, 83syl5bi 231 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ∅ ∈ 𝐿))
8534, 84mtod 188 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
86 df-nel 2783 . . . 4 (∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
8785, 86sylibr 223 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)))
88 vex 3176 . . . . . . . . 9 𝑟 ∈ V
8911elrnmpt 5293 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥)))
9088, 89ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥))
91 imaeq2 5381 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
9291eqeq2d 2620 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑟 = (𝐹𝑥) ↔ 𝑟 = (𝐹𝑢)))
9392cbvrexv 3148 . . . . . . . 8 (∃𝑥𝐿 𝑟 = (𝐹𝑥) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
9490, 93bitri 263 . . . . . . 7 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
95 vex 3176 . . . . . . . . 9 𝑠 ∈ V
9611elrnmpt 5293 . . . . . . . . 9 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
9795, 96ax-mp 5 . . . . . . . 8 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
98 imaeq2 5381 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
9998eqeq2d 2620 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑣)))
10099cbvrexv 3148 . . . . . . . 8 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10197, 100bitri 263 . . . . . . 7 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10294, 101anbi12i 729 . . . . . 6 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
103 reeanv 3086 . . . . . 6 (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
104102, 103bitr4i 266 . . . . 5 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))
105 filin 21468 . . . . . . . . . . . . . 14 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑢𝐿𝑣𝐿) → (𝑢𝑣) ∈ 𝐿)
1061053expb 1258 . . . . . . . . . . . . 13 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
107106adantlr 747 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
108 eqidd 2611 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣)))
109 imaeq2 5381 . . . . . . . . . . . . . 14 (𝑥 = (𝑢𝑣) → (𝐹𝑥) = (𝐹 “ (𝑢𝑣)))
110109eqeq2d 2620 . . . . . . . . . . . . 13 (𝑥 = (𝑢𝑣) → ((𝐹 “ (𝑢𝑣)) = (𝐹𝑥) ↔ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))))
111110rspcev 3282 . . . . . . . . . . . 12 (((𝑢𝑣) ∈ 𝐿 ∧ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
112107, 108, 111syl2anc 691 . . . . . . . . . . 11 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
1131123adantl1 1210 . . . . . . . . . 10 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
114113ad2ant2r 779 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
115 simpll1 1093 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑌𝐴)
116 cnvimass 5404 . . . . . . . . . . . . . 14 (𝐹 “ (𝑢𝑣)) ⊆ dom 𝐹
117116, 3syl5sseq 3616 . . . . . . . . . . . . 13 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
1181173ad2ant3 1077 . . . . . . . . . . . 12 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
119118ad2antrr 758 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
120115, 119ssexd 4733 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ V)
12111elrnmpt 5293 . . . . . . . . . 10 ((𝐹 “ (𝑢𝑣)) ∈ V → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
122120, 121syl 17 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
123114, 122mpbird 246 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
124 simprrl 800 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑟 = (𝐹𝑢))
125 simprrr 801 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑠 = (𝐹𝑣))
126124, 125ineq12d 3777 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
127 funcnvcnv 5870 . . . . . . . . . . . . 13 (Fun 𝐹 → Fun 𝐹)
128 imain 5888 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
12947, 127, 1283syl 18 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
1301293ad2ant3 1077 . . . . . . . . . . 11 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
131130ad2antrr 758 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
132126, 131eqtr4d 2647 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = (𝐹 “ (𝑢𝑣)))
133 eqimss2 3621 . . . . . . . . 9 ((𝑟𝑠) = (𝐹 “ (𝑢𝑣)) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
134132, 133syl 17 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
135 sseq1 3589 . . . . . . . . 9 (𝑡 = (𝐹 “ (𝑢𝑣)) → (𝑡 ⊆ (𝑟𝑠) ↔ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)))
136135rspcev 3282 . . . . . . . 8 (((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
137123, 134, 136syl2anc 691 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
138137exp32 629 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑢𝐿𝑣𝐿) → ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))))
139138rexlimdvv 3019 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
140104, 139syl5bi 231 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
141140ralrimivv 2953 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
14231, 87, 1413jca 1235 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
143 isfbas2 21449 . . 3 (𝑌𝐴 → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
1446, 143syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
14514, 142, 144mpbir2and 959 1 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  fBascfbas 19555  Filcfil 21459 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-nel 2783  df-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-fbas 19564  df-fil 21460 This theorem is referenced by:  rnelfm  21567  fmfnfm  21572
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