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Theorem qtoprest 21800
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 6300 . . . . . . 7 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . . . . 6 (𝜑𝐹 Fn 𝑋)
5 qtopid 21788 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 579 . . . . 5 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 qtoprest.5 . . . . . . 7 (𝜑𝐴 = (𝐹𝑈))
8 cnvimass 5667 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
9 fndm 6168 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
104, 9syl 17 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝑋)
118, 10syl5sseq 3813 . . . . . . 7 (𝜑 → (𝐹𝑈) ⊆ 𝑋)
127, 11eqsstrd 3799 . . . . . 6 (𝜑𝐴𝑋)
13 toponuni 20998 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
141, 13syl 17 . . . . . 6 (𝜑𝑋 = 𝐽)
1512, 14sseqtrd 3801 . . . . 5 (𝜑𝐴 𝐽)
16 eqid 2765 . . . . . 6 𝐽 = 𝐽
1716cnrest 21369 . . . . 5 ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 𝐽) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
186, 15, 17syl2anc 579 . . . 4 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
19 qtoptopon 21787 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
201, 2, 19syl2anc 579 . . . . 5 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
21 df-ima 5290 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
227imaeq2d 5648 . . . . . . . 8 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝐹𝑈)))
23 qtoprest.4 . . . . . . . . 9 (𝜑𝑈𝑌)
24 foimacnv 6337 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑈𝑌) → (𝐹 “ (𝐹𝑈)) = 𝑈)
252, 23, 24syl2anc 579 . . . . . . . 8 (𝜑 → (𝐹 “ (𝐹𝑈)) = 𝑈)
2622, 25eqtrd 2799 . . . . . . 7 (𝜑 → (𝐹𝐴) = 𝑈)
2721, 26syl5eqr 2813 . . . . . 6 (𝜑 → ran (𝐹𝐴) = 𝑈)
28 eqimss 3817 . . . . . 6 (ran (𝐹𝐴) = 𝑈 → ran (𝐹𝐴) ⊆ 𝑈)
2927, 28syl 17 . . . . 5 (𝜑 → ran (𝐹𝐴) ⊆ 𝑈)
30 cnrest2 21370 . . . . 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 1490 . . . 4 (𝜑 → ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
3218, 31mpbid 223 . . 3 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
33 resttopon 21245 . . . 4 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
3420, 23, 33syl2anc 579 . . 3 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
35 qtopss 21798 . . 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 1490 . 2 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽t 𝐴) qTop (𝐹𝐴)))
37 resttopon 21245 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
381, 12, 37syl2anc 579 . . . . 5 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
39 fnfun 6166 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
404, 39syl 17 . . . . . . 7 (𝜑 → Fun 𝐹)
4112, 10sseqtr4d 3802 . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐹)
42 fores 6307 . . . . . . 7 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
4340, 41, 42syl2anc 579 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴onto→(𝐹𝐴))
44 foeq3 6296 . . . . . . 7 ((𝐹𝐴) = 𝑈 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4526, 44syl 17 . . . . . 6 (𝜑 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4643, 45mpbid 223 . . . . 5 (𝜑 → (𝐹𝐴):𝐴onto𝑈)
47 elqtop3 21786 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹𝐴):𝐴onto𝑈) → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
4838, 46, 47syl2anc 579 . . . 4 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
49 cnvresima 5809 . . . . . . . 8 ((𝐹𝐴) “ 𝑥) = ((𝐹𝑥) ∩ 𝐴)
50 imass2 5683 . . . . . . . . . . 11 (𝑥𝑈 → (𝐹𝑥) ⊆ (𝐹𝑈))
5150adantl 473 . . . . . . . . . 10 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹𝑈))
527adantr 472 . . . . . . . . . 10 ((𝜑𝑥𝑈) → 𝐴 = (𝐹𝑈))
5351, 52sseqtr4d 3802 . . . . . . . . 9 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ 𝐴)
54 df-ss 3746 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝐴 ↔ ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5553, 54sylib 209 . . . . . . . 8 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5649, 55syl5eq 2811 . . . . . . 7 ((𝜑𝑥𝑈) → ((𝐹𝐴) “ 𝑥) = (𝐹𝑥))
5756eleq1d 2829 . . . . . 6 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) ↔ (𝐹𝑥) ∈ (𝐽t 𝐴)))
58 simplrl 795 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑈)
59 df-ss 3746 . . . . . . . . . 10 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
6058, 59sylib 209 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) = 𝑥)
61 topontop 20997 . . . . . . . . . . . 12 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
6220, 61syl 17 . . . . . . . . . . 11 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
6362ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐽 qTop 𝐹) ∈ Top)
64 toponmax 21010 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
651, 64syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
66 fornex 7333 . . . . . . . . . . . . 13 (𝑋𝐽 → (𝐹:𝑋onto𝑌𝑌 ∈ V))
6765, 2, 66sylc 65 . . . . . . . . . . . 12 (𝜑𝑌 ∈ V)
6867, 23ssexd 4966 . . . . . . . . . . 11 (𝜑𝑈 ∈ V)
6968ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈 ∈ V)
7023ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈𝑌)
7158, 70sstrd 3771 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑌)
72 topontop 20997 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
731, 72syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ Top)
74 restopn2 21261 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7573, 74sylan 575 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7675simprbda 492 . . . . . . . . . . . . 13 (((𝜑𝐴𝐽) ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → (𝐹𝑥) ∈ 𝐽)
7776adantrl 707 . . . . . . . . . . . 12 (((𝜑𝐴𝐽) ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐹𝑥) ∈ 𝐽)
7877an32s 642 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐹𝑥) ∈ 𝐽)
79 elqtop3 21786 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
801, 2, 79syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8180ad2antrr 717 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8271, 78, 81mpbir2and 704 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
83 elrestr 16357 . . . . . . . . . 10 (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8463, 69, 82, 83syl3anc 1490 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8560, 84eqeltrrd 2845 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8634ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
87 toponuni 20998 . . . . . . . . . . . 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 3889 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) = ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
9023ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈𝑌)
9120ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
92 toponuni 20998 . . . . . . . . . . . . 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 3801 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 (𝐽 qTop 𝐹))
9590ssdifssd 3910 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑌)
9640ad2antrr 717 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
97 funcnvcnv 6134 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun 𝐹)
98 imadif 6151 . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐹𝑈))
101100difeq1d 3889 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
10299, 101eqtr4d 2802 . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
105 toponuni 20998 . . . . . . . . . . . . . . . . 17 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = (𝐽t 𝐴))
106104, 105syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐽t 𝐴))
107106difeq1d 3889 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ( (𝐽t 𝐴) ∖ (𝐹𝑥)))
108 topontop 20997 . . . . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (𝐽t 𝐴))
111 eqid 2765 . . . . . . . . . . . . . . . . 17 (𝐽t 𝐴) = (𝐽t 𝐴)
112111opncld 21117 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐴) ∈ Top ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
113109, 110, 112syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
114107, 113eqeltrd 2844 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
115 restcldr 21258 . . . . . . . . . . . . . 14 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴))) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
116103, 114, 115syl2anc 579 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
117102, 116eqeltrd 2844 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))
118 qtopcld 21796 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
1191, 2, 118syl2anc 579 . . . . . . . . . . . . 13 (𝜑 → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
120119ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
12195, 117, 120mpbir2and 704 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
122 difssd 3900 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑈)
123 eqid 2765 . . . . . . . . . . . 12 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
124123restcldi 21257 . . . . . . . . . . 11 ((𝑈 (𝐽 qTop 𝐹) ∧ (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ∧ (𝑈𝑥) ⊆ 𝑈) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12594, 121, 122, 124syl3anc 1490 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12689, 125eqeltrrd 2845 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
127 topontop 20997 . . . . . . . . . . 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 795 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥𝑈)
130129, 88sseqtrd 3801 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ((𝐽 qTop 𝐹) ↾t 𝑈))
131 eqid 2765 . . . . . . . . . . 11 ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 qTop 𝐹) ↾t 𝑈)
132131isopn2 21116 . . . . . . . . . 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 579 . . . . . . . . 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 248 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
135 qtoprest.6 . . . . . . . . 9 (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
136135adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
13785, 134, 136mpjaodan 981 . . . . . . 7 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
138137expr 448 . . . . . 6 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
13957, 138sylbid 231 . . . . 5 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140139expimpd 445 . . . 4 (𝜑 → ((𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
14148, 140sylbid 231 . . 3 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
142141ssrdv 3767 . 2 (𝜑 → ((𝐽t 𝐴) qTop (𝐹𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
14336, 142eqssd 3778 1 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  cin 3731  wss 3732   cuni 4594  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Fun wfun 6062   Fn wfn 6063  ontowfo 6066  cfv 6068  (class class class)co 6842  t crest 16349   qTop cqtop 16431  Topctop 20977  TopOnctopon 20994  Clsdccld 21100   Cn ccn 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-fin 8164  df-fi 8524  df-rest 16351  df-topgen 16372  df-qtop 16435  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-cn 21311
This theorem is referenced by: (None)
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