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Theorem qtoprest 21439
Description: If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (𝐹𝑈) 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 6079 . . . . . . 7 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . . . . 6 (𝜑𝐹 Fn 𝑋)
5 qtopid 21427 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 692 . . . . 5 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 qtoprest.5 . . . . . . 7 (𝜑𝐴 = (𝐹𝑈))
8 cnvimass 5449 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
9 fndm 5953 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
104, 9syl 17 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝑋)
118, 10syl5sseq 3637 . . . . . . 7 (𝜑 → (𝐹𝑈) ⊆ 𝑋)
127, 11eqsstrd 3623 . . . . . 6 (𝜑𝐴𝑋)
13 toponuni 20647 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
141, 13syl 17 . . . . . 6 (𝜑𝑋 = 𝐽)
1512, 14sseqtrd 3625 . . . . 5 (𝜑𝐴 𝐽)
16 eqid 2621 . . . . . 6 𝐽 = 𝐽
1716cnrest 21008 . . . . 5 ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 𝐽) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
186, 15, 17syl2anc 692 . . . 4 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
19 qtoptopon 21426 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
201, 2, 19syl2anc 692 . . . . 5 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
21 df-ima 5092 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
227imaeq2d 5430 . . . . . . . 8 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝐹𝑈)))
23 qtoprest.4 . . . . . . . . 9 (𝜑𝑈𝑌)
24 foimacnv 6116 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑈𝑌) → (𝐹 “ (𝐹𝑈)) = 𝑈)
252, 23, 24syl2anc 692 . . . . . . . 8 (𝜑 → (𝐹 “ (𝐹𝑈)) = 𝑈)
2622, 25eqtrd 2655 . . . . . . 7 (𝜑 → (𝐹𝐴) = 𝑈)
2721, 26syl5eqr 2669 . . . . . 6 (𝜑 → ran (𝐹𝐴) = 𝑈)
28 eqimss 3641 . . . . . 6 (ran (𝐹𝐴) = 𝑈 → ran (𝐹𝐴) ⊆ 𝑈)
2927, 28syl 17 . . . . 5 (𝜑 → ran (𝐹𝐴) ⊆ 𝑈)
30 cnrest2 21009 . . . . 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 1323 . . . 4 (𝜑 → ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
3218, 31mpbid 222 . . 3 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
33 resttopon 20884 . . . 4 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
3420, 23, 33syl2anc 692 . . 3 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
35 qtopss 21437 . . 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 1323 . 2 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽t 𝐴) qTop (𝐹𝐴)))
37 resttopon 20884 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
381, 12, 37syl2anc 692 . . . . 5 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
39 fnfun 5951 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
404, 39syl 17 . . . . . . 7 (𝜑 → Fun 𝐹)
4112, 10sseqtr4d 3626 . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐹)
42 fores 6086 . . . . . . 7 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
4340, 41, 42syl2anc 692 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴onto→(𝐹𝐴))
44 foeq3 6075 . . . . . . 7 ((𝐹𝐴) = 𝑈 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4526, 44syl 17 . . . . . 6 (𝜑 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4643, 45mpbid 222 . . . . 5 (𝜑 → (𝐹𝐴):𝐴onto𝑈)
47 elqtop3 21425 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹𝐴):𝐴onto𝑈) → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
4838, 46, 47syl2anc 692 . . . 4 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
49 cnvresima 5587 . . . . . . . 8 ((𝐹𝐴) “ 𝑥) = ((𝐹𝑥) ∩ 𝐴)
50 imass2 5465 . . . . . . . . . . 11 (𝑥𝑈 → (𝐹𝑥) ⊆ (𝐹𝑈))
5150adantl 482 . . . . . . . . . 10 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹𝑈))
527adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑈) → 𝐴 = (𝐹𝑈))
5351, 52sseqtr4d 3626 . . . . . . . . 9 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ 𝐴)
54 df-ss 3573 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝐴 ↔ ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5553, 54sylib 208 . . . . . . . 8 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5649, 55syl5eq 2667 . . . . . . 7 ((𝜑𝑥𝑈) → ((𝐹𝐴) “ 𝑥) = (𝐹𝑥))
5756eleq1d 2683 . . . . . 6 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) ↔ (𝐹𝑥) ∈ (𝐽t 𝐴)))
58 simplrl 799 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑈)
59 df-ss 3573 . . . . . . . . . 10 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
6058, 59sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) = 𝑥)
61 topontop 20646 . . . . . . . . . . . 12 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
6220, 61syl 17 . . . . . . . . . . 11 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
6362ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐽 qTop 𝐹) ∈ Top)
64 toponmax 20648 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
651, 64syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
66 fornex 7089 . . . . . . . . . . . . 13 (𝑋𝐽 → (𝐹:𝑋onto𝑌𝑌 ∈ V))
6765, 2, 66sylc 65 . . . . . . . . . . . 12 (𝜑𝑌 ∈ V)
6867, 23ssexd 4770 . . . . . . . . . . 11 (𝜑𝑈 ∈ V)
6968ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈 ∈ V)
7023ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈𝑌)
7158, 70sstrd 3597 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑌)
72 topontop 20646 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
731, 72syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ Top)
74 restopn2 20900 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7573, 74sylan 488 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7675simprbda 652 . . . . . . . . . . . . 13 (((𝜑𝐴𝐽) ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → (𝐹𝑥) ∈ 𝐽)
7776adantrl 751 . . . . . . . . . . . 12 (((𝜑𝐴𝐽) ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐹𝑥) ∈ 𝐽)
7877an32s 845 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐹𝑥) ∈ 𝐽)
79 elqtop3 21425 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
801, 2, 79syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8180ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8271, 78, 81mpbir2and 956 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
83 elrestr 16017 . . . . . . . . . 10 (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8463, 69, 82, 83syl3anc 1323 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8560, 84eqeltrrd 2699 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8634ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
87 toponuni 20647 . . . . . . . . . . . 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 3710 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) = ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
9023ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈𝑌)
9120ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
92 toponuni 20647 . . . . . . . . . . . . 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 3625 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 (𝐽 qTop 𝐹))
9590ssdifssd 3731 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑌)
9640ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
97 funcnvcnv 5919 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun 𝐹)
98 imadif 5936 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐹𝑈))
101100difeq1d 3710 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
10299, 101eqtr4d 2658 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) = (𝐴 ∖ (𝐹𝑥)))
103 simpr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
10438ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
105 toponuni 20647 . . . . . . . . . . . . . . . . 17 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = (𝐽t 𝐴))
106104, 105syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐽t 𝐴))
107106difeq1d 3710 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ( (𝐽t 𝐴) ∖ (𝐹𝑥)))
108 topontop 20646 . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (𝐽t 𝐴))
111 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝐽t 𝐴) = (𝐽t 𝐴)
112111opncld 20756 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐴) ∈ Top ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
113109, 110, 112syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
114107, 113eqeltrd 2698 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
115 restcldr 20897 . . . . . . . . . . . . . 14 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴))) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
116103, 114, 115syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
117102, 116eqeltrd 2698 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))
118 qtopcld 21435 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
1191, 2, 118syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
120119ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
12195, 117, 120mpbir2and 956 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
122 difssd 3721 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑈)
123 eqid 2621 . . . . . . . . . . . 12 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
124123restcldi 20896 . . . . . . . . . . 11 ((𝑈 (𝐽 qTop 𝐹) ∧ (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ∧ (𝑈𝑥) ⊆ 𝑈) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12594, 121, 122, 124syl3anc 1323 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12689, 125eqeltrrd 2699 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
127 topontop 20646 . . . . . . . . . . 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 799 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥𝑈)
130129, 88sseqtrd 3625 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ((𝐽 qTop 𝐹) ↾t 𝑈))
131 eqid 2621 . . . . . . . . . . 11 ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 qTop 𝐹) ↾t 𝑈)
132131isopn2 20755 . . . . . . . . . 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 692 . . . . . . . . 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 247 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
135 qtoprest.6 . . . . . . . . 9 (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
136135adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
13785, 134, 136mpjaodan 826 . . . . . . 7 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
138137expr 642 . . . . . 6 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
13957, 138sylbid 230 . . . . 5 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140139expimpd 628 . . . 4 (𝜑 → ((𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
14148, 140sylbid 230 . . 3 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
142141ssrdv 3593 . 2 (𝜑 → ((𝐽t 𝐴) qTop (𝐹𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
14336, 142eqssd 3604 1 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  cdif 3556  cin 3558  wss 3559   cuni 4407  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Fun wfun 5846   Fn wfn 5847  ontowfo 5850  cfv 5852  (class class class)co 6610  t crest 16009   qTop cqtop 16091  Topctop 20626  TopOnctopon 20643  Clsdccld 20739   Cn ccn 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-fin 7910  df-fi 8268  df-rest 16011  df-topgen 16032  df-qtop 16095  df-top 20627  df-topon 20644  df-bases 20670  df-cld 20742  df-cn 20950
This theorem is referenced by: (None)
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