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Theorem qtoprest 21330
 Description: If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (◡𝐹 “ 𝑈) for some 𝑈), then the restriction of the quotient map 𝐹 to 𝐴 is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
qtoprest.2 (𝜑𝐽 ∈ (TopOn‘𝑋))
qtoprest.3 (𝜑𝐹:𝑋onto𝑌)
qtoprest.4 (𝜑𝑈𝑌)
qtoprest.5 (𝜑𝐴 = (𝐹𝑈))
qtoprest.6 (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
Assertion
Ref Expression
qtoprest (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))

Proof of Theorem qtoprest
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 qtoprest.2 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 qtoprest.3 . . . . . . 7 (𝜑𝐹:𝑋onto𝑌)
3 fofn 6030 . . . . . . 7 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . . . . 6 (𝜑𝐹 Fn 𝑋)
5 qtopid 21318 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 691 . . . . 5 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 qtoprest.5 . . . . . . 7 (𝜑𝐴 = (𝐹𝑈))
8 cnvimass 5404 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
9 fndm 5904 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
104, 9syl 17 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝑋)
118, 10syl5sseq 3616 . . . . . . 7 (𝜑 → (𝐹𝑈) ⊆ 𝑋)
127, 11eqsstrd 3602 . . . . . 6 (𝜑𝐴𝑋)
13 toponuni 20542 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
141, 13syl 17 . . . . . 6 (𝜑𝑋 = 𝐽)
1512, 14sseqtrd 3604 . . . . 5 (𝜑𝐴 𝐽)
16 eqid 2610 . . . . . 6 𝐽 = 𝐽
1716cnrest 20899 . . . . 5 ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 𝐽) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
186, 15, 17syl2anc 691 . . . 4 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
19 qtoptopon 21317 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
201, 2, 19syl2anc 691 . . . . 5 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
21 df-ima 5051 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
227imaeq2d 5385 . . . . . . . 8 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝐹𝑈)))
23 qtoprest.4 . . . . . . . . 9 (𝜑𝑈𝑌)
24 foimacnv 6067 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑈𝑌) → (𝐹 “ (𝐹𝑈)) = 𝑈)
252, 23, 24syl2anc 691 . . . . . . . 8 (𝜑 → (𝐹 “ (𝐹𝑈)) = 𝑈)
2622, 25eqtrd 2644 . . . . . . 7 (𝜑 → (𝐹𝐴) = 𝑈)
2721, 26syl5eqr 2658 . . . . . 6 (𝜑 → ran (𝐹𝐴) = 𝑈)
28 eqimss 3620 . . . . . 6 (ran (𝐹𝐴) = 𝑈 → ran (𝐹𝐴) ⊆ 𝑈)
2927, 28syl 17 . . . . 5 (𝜑 → ran (𝐹𝐴) ⊆ 𝑈)
30 cnrest2 20900 . . . . 5 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ ran (𝐹𝐴) ⊆ 𝑈𝑈𝑌) → ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
3120, 29, 23, 30syl3anc 1318 . . . 4 (𝜑 → ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
3218, 31mpbid 221 . . 3 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
33 resttopon 20775 . . . 4 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
3420, 23, 33syl2anc 691 . . 3 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
35 qtopss 21328 . . 3 (((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)) ∧ ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) ∧ ran (𝐹𝐴) = 𝑈) → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽t 𝐴) qTop (𝐹𝐴)))
3632, 34, 27, 35syl3anc 1318 . 2 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽t 𝐴) qTop (𝐹𝐴)))
37 resttopon 20775 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
381, 12, 37syl2anc 691 . . . . 5 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
39 fnfun 5902 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
404, 39syl 17 . . . . . . 7 (𝜑 → Fun 𝐹)
4112, 10sseqtr4d 3605 . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐹)
42 fores 6037 . . . . . . 7 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
4340, 41, 42syl2anc 691 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴onto→(𝐹𝐴))
44 foeq3 6026 . . . . . . 7 ((𝐹𝐴) = 𝑈 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4526, 44syl 17 . . . . . 6 (𝜑 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4643, 45mpbid 221 . . . . 5 (𝜑 → (𝐹𝐴):𝐴onto𝑈)
47 elqtop3 21316 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹𝐴):𝐴onto𝑈) → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
4838, 46, 47syl2anc 691 . . . 4 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
49 cnvresima 5541 . . . . . . . 8 ((𝐹𝐴) “ 𝑥) = ((𝐹𝑥) ∩ 𝐴)
50 imass2 5420 . . . . . . . . . . 11 (𝑥𝑈 → (𝐹𝑥) ⊆ (𝐹𝑈))
5150adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹𝑈))
527adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑈) → 𝐴 = (𝐹𝑈))
5351, 52sseqtr4d 3605 . . . . . . . . 9 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ 𝐴)
54 df-ss 3554 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝐴 ↔ ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5553, 54sylib 207 . . . . . . . 8 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5649, 55syl5eq 2656 . . . . . . 7 ((𝜑𝑥𝑈) → ((𝐹𝐴) “ 𝑥) = (𝐹𝑥))
5756eleq1d 2672 . . . . . 6 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) ↔ (𝐹𝑥) ∈ (𝐽t 𝐴)))
58 simplrl 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑈)
59 df-ss 3554 . . . . . . . . . 10 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
6058, 59sylib 207 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) = 𝑥)
61 topontop 20541 . . . . . . . . . . . 12 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
6220, 61syl 17 . . . . . . . . . . 11 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
6362ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐽 qTop 𝐹) ∈ Top)
64 toponmax 20543 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
651, 64syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
66 fornex 7028 . . . . . . . . . . . . 13 (𝑋𝐽 → (𝐹:𝑋onto𝑌𝑌 ∈ V))
6765, 2, 66sylc 63 . . . . . . . . . . . 12 (𝜑𝑌 ∈ V)
6867, 23ssexd 4733 . . . . . . . . . . 11 (𝜑𝑈 ∈ V)
6968ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈 ∈ V)
7023ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈𝑌)
7158, 70sstrd 3578 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑌)
72 topontop 20541 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
731, 72syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ Top)
74 restopn2 20791 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7573, 74sylan 487 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7675simprbda 651 . . . . . . . . . . . . 13 (((𝜑𝐴𝐽) ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → (𝐹𝑥) ∈ 𝐽)
7776adantrl 748 . . . . . . . . . . . 12 (((𝜑𝐴𝐽) ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐹𝑥) ∈ 𝐽)
7877an32s 842 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐹𝑥) ∈ 𝐽)
79 elqtop3 21316 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
801, 2, 79syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8180ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8271, 78, 81mpbir2and 959 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
83 elrestr 15912 . . . . . . . . . 10 (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8463, 69, 82, 83syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8560, 84eqeltrrd 2689 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8634ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
87 toponuni 20542 . . . . . . . . . . . 12 (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → 𝑈 = ((𝐽 qTop 𝐹) ↾t 𝑈))
8886, 87syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 = ((𝐽 qTop 𝐹) ↾t 𝑈))
8988difeq1d 3689 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) = ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
9023ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈𝑌)
9120ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
92 toponuni 20542 . . . . . . . . . . . . 13 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
9391, 92syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑌 = (𝐽 qTop 𝐹))
9490, 93sseqtrd 3604 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 (𝐽 qTop 𝐹))
9590ssdifssd 3710 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑌)
9640ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
97 funcnvcnv 5870 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun 𝐹)
98 imadif 5887 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (𝐹 “ (𝑈𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
9996, 97, 983syl 18 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
1007ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐹𝑈))
101100difeq1d 3689 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
10299, 101eqtr4d 2647 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) = (𝐴 ∖ (𝐹𝑥)))
103 simpr 476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
10438ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
105 toponuni 20542 . . . . . . . . . . . . . . . . 17 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = (𝐽t 𝐴))
106104, 105syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐽t 𝐴))
107106difeq1d 3689 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ( (𝐽t 𝐴) ∖ (𝐹𝑥)))
108 topontop 20541 . . . . . . . . . . . . . . . . 17 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → (𝐽t 𝐴) ∈ Top)
109104, 108syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
110 simplrr 797 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (𝐽t 𝐴))
111 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝐽t 𝐴) = (𝐽t 𝐴)
112111opncld 20647 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐴) ∈ Top ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
113109, 110, 112syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
114107, 113eqeltrd 2688 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
115 restcldr 20788 . . . . . . . . . . . . . 14 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴))) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
116103, 114, 115syl2anc 691 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
117102, 116eqeltrd 2688 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))
118 qtopcld 21326 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
1191, 2, 118syl2anc 691 . . . . . . . . . . . . 13 (𝜑 → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
120119ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
12195, 117, 120mpbir2and 959 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
122 difssd 3700 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑈)
123 eqid 2610 . . . . . . . . . . . 12 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
124123restcldi 20787 . . . . . . . . . . 11 ((𝑈 (𝐽 qTop 𝐹) ∧ (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ∧ (𝑈𝑥) ⊆ 𝑈) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12594, 121, 122, 124syl3anc 1318 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12689, 125eqeltrrd 2689 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
127 topontop 20541 . . . . . . . . . . 11 (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
12886, 127syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
129 simplrl 796 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥𝑈)
130129, 88sseqtrd 3604 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ((𝐽 qTop 𝐹) ↾t 𝑈))
131 eqid 2610 . . . . . . . . . . 11 ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 qTop 𝐹) ↾t 𝑈)
132131isopn2 20646 . . . . . . . . . 10 ((((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top ∧ 𝑥 ((𝐽 qTop 𝐹) ↾t 𝑈)) → (𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈) ↔ ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈))))
133128, 130, 132syl2anc 691 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈) ↔ ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈))))
134126, 133mpbird 246 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
135 qtoprest.6 . . . . . . . . 9 (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
136135adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
13785, 134, 136mpjaodan 823 . . . . . . 7 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
138137expr 641 . . . . . 6 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
13957, 138sylbid 229 . . . . 5 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140139expimpd 627 . . . 4 (𝜑 → ((𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
14148, 140sylbid 229 . . 3 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
142141ssrdv 3574 . 2 (𝜑 → ((𝐽t 𝐴) qTop (𝐹𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
14336, 142eqssd 3585 1 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838 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 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-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-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-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-qtop 15990  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cn 20841 This theorem is referenced by: (None)
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