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Theorem qtoprest 22009
 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 6460 . . . . . . 7 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . . . . 6 (𝜑𝐹 Fn 𝑋)
5 qtopid 21997 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 584 . . . . 5 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 qtoprest.5 . . . . . . 7 (𝜑𝐴 = (𝐹𝑈))
8 cnvimass 5825 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
9 fndm 6325 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
104, 9syl 17 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝑋)
118, 10sseqtrid 3940 . . . . . . 7 (𝜑 → (𝐹𝑈) ⊆ 𝑋)
127, 11eqsstrd 3926 . . . . . 6 (𝜑𝐴𝑋)
13 toponuni 21206 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
141, 13syl 17 . . . . . 6 (𝜑𝑋 = 𝐽)
1512, 14sseqtrd 3928 . . . . 5 (𝜑𝐴 𝐽)
16 eqid 2795 . . . . . 6 𝐽 = 𝐽
1716cnrest 21577 . . . . 5 ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 𝐽) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
186, 15, 17syl2anc 584 . . . 4 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)))
19 qtoptopon 21996 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
201, 2, 19syl2anc 584 . . . . 5 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
21 df-ima 5456 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
227imaeq2d 5806 . . . . . . . 8 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝐹𝑈)))
23 qtoprest.4 . . . . . . . . 9 (𝜑𝑈𝑌)
24 foimacnv 6500 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑈𝑌) → (𝐹 “ (𝐹𝑈)) = 𝑈)
252, 23, 24syl2anc 584 . . . . . . . 8 (𝜑 → (𝐹 “ (𝐹𝑈)) = 𝑈)
2622, 25eqtrd 2831 . . . . . . 7 (𝜑 → (𝐹𝐴) = 𝑈)
2721, 26syl5eqr 2845 . . . . . 6 (𝜑 → ran (𝐹𝐴) = 𝑈)
28 eqimss 3944 . . . . . 6 (ran (𝐹𝐴) = 𝑈 → ran (𝐹𝐴) ⊆ 𝑈)
2927, 28syl 17 . . . . 5 (𝜑 → ran (𝐹𝐴) ⊆ 𝑈)
30 cnrest2 21578 . . . . 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 1364 . . . 4 (𝜑 → ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
3218, 31mpbid 233 . . 3 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
33 resttopon 21453 . . . 4 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
3420, 23, 33syl2anc 584 . . 3 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
35 qtopss 22007 . . 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 1364 . 2 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽t 𝐴) qTop (𝐹𝐴)))
37 resttopon 21453 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
381, 12, 37syl2anc 584 . . . . 5 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
39 fnfun 6323 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
404, 39syl 17 . . . . . . 7 (𝜑 → Fun 𝐹)
4112, 10sseqtr4d 3929 . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐹)
42 fores 6468 . . . . . . 7 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
4340, 41, 42syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴onto→(𝐹𝐴))
44 foeq3 6456 . . . . . . 7 ((𝐹𝐴) = 𝑈 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4526, 44syl 17 . . . . . 6 (𝜑 → ((𝐹𝐴):𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto𝑈))
4643, 45mpbid 233 . . . . 5 (𝜑 → (𝐹𝐴):𝐴onto𝑈)
47 elqtop3 21995 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹𝐴):𝐴onto𝑈) → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
4838, 46, 47syl2anc 584 . . . 4 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) ↔ (𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴))))
49 cnvresima 5962 . . . . . . . 8 ((𝐹𝐴) “ 𝑥) = ((𝐹𝑥) ∩ 𝐴)
50 imass2 5841 . . . . . . . . . . 11 (𝑥𝑈 → (𝐹𝑥) ⊆ (𝐹𝑈))
5150adantl 482 . . . . . . . . . 10 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹𝑈))
527adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑈) → 𝐴 = (𝐹𝑈))
5351, 52sseqtr4d 3929 . . . . . . . . 9 ((𝜑𝑥𝑈) → (𝐹𝑥) ⊆ 𝐴)
54 df-ss 3874 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝐴 ↔ ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5553, 54sylib 219 . . . . . . . 8 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∩ 𝐴) = (𝐹𝑥))
5649, 55syl5eq 2843 . . . . . . 7 ((𝜑𝑥𝑈) → ((𝐹𝐴) “ 𝑥) = (𝐹𝑥))
5756eleq1d 2867 . . . . . 6 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) ↔ (𝐹𝑥) ∈ (𝐽t 𝐴)))
58 simplrl 773 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑈)
59 df-ss 3874 . . . . . . . . . 10 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
6058, 59sylib 219 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) = 𝑥)
61 topontop 21205 . . . . . . . . . . . 12 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
6220, 61syl 17 . . . . . . . . . . 11 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
6362ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐽 qTop 𝐹) ∈ Top)
64 toponmax 21218 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
651, 64syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
66 fornex 7513 . . . . . . . . . . . . 13 (𝑋𝐽 → (𝐹:𝑋onto𝑌𝑌 ∈ V))
6765, 2, 66sylc 65 . . . . . . . . . . . 12 (𝜑𝑌 ∈ V)
6867, 23ssexd 5119 . . . . . . . . . . 11 (𝜑𝑈 ∈ V)
6968ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈 ∈ V)
7023ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑈𝑌)
7158, 70sstrd 3899 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥𝑌)
72 topontop 21205 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
731, 72syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ Top)
74 restopn2 21469 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7573, 74sylan 580 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐽) → ((𝐹𝑥) ∈ (𝐽t 𝐴) ↔ ((𝐹𝑥) ∈ 𝐽 ∧ (𝐹𝑥) ⊆ 𝐴)))
7675simprbda 499 . . . . . . . . . . . . 13 (((𝜑𝐴𝐽) ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → (𝐹𝑥) ∈ 𝐽)
7776adantrl 712 . . . . . . . . . . . 12 (((𝜑𝐴𝐽) ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → (𝐹𝑥) ∈ 𝐽)
7877an32s 648 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝐹𝑥) ∈ 𝐽)
79 elqtop3 21995 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
801, 2, 79syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8180ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
8271, 78, 81mpbir2and 709 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
83 elrestr 16531 . . . . . . . . . 10 (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8463, 69, 82, 83syl3anc 1364 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → (𝑥𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8560, 84eqeltrrd 2884 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
8634ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
87 toponuni 21206 . . . . . . . . . . . 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 4019 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) = ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
9023ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈𝑌)
9120ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
92 toponuni 21206 . . . . . . . . . . . . 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 3928 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 (𝐽 qTop 𝐹))
9590ssdifssd 4040 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑌)
9640ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
97 funcnvcnv 6291 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun 𝐹)
98 imadif 6308 . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐹𝑈))
101100difeq1d 4019 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ((𝐹𝑈) ∖ (𝐹𝑥)))
10299, 101eqtr4d 2834 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) = (𝐴 ∖ (𝐹𝑥)))
103 simpr 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
10438ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
105 toponuni 21206 . . . . . . . . . . . . . . . . 17 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = (𝐽t 𝐴))
106104, 105syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (𝐽t 𝐴))
107106difeq1d 4019 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) = ( (𝐽t 𝐴) ∖ (𝐹𝑥)))
108 topontop 21205 . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (𝐽t 𝐴))
111 eqid 2795 . . . . . . . . . . . . . . . . 17 (𝐽t 𝐴) = (𝐽t 𝐴)
112111opncld 21325 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐴) ∈ Top ∧ (𝐹𝑥) ∈ (𝐽t 𝐴)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
113109, 110, 112syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( (𝐽t 𝐴) ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
114107, 113eqeltrd 2883 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴)))
115 restcldr 21466 . . . . . . . . . . . . . 14 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘(𝐽t 𝐴))) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
116103, 114, 115syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
117102, 116eqeltrd 2883 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))
118 qtopcld 22005 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
1191, 2, 118syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
120119ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝑈𝑥)) ∈ (Clsd‘𝐽))))
12195, 117, 120mpbir2and 709 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
122 difssd 4030 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ⊆ 𝑈)
123 eqid 2795 . . . . . . . . . . . 12 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
124123restcldi 21465 . . . . . . . . . . 11 ((𝑈 (𝐽 qTop 𝐹) ∧ (𝑈𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ∧ (𝑈𝑥) ⊆ 𝑈) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12594, 121, 122, 124syl3anc 1364 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
12689, 125eqeltrrd 2884 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ( ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
127 topontop 21205 . . . . . . . . . . 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 773 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥𝑈)
130129, 88sseqtrd 3928 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ((𝐽 qTop 𝐹) ↾t 𝑈))
131 eqid 2795 . . . . . . . . . . 11 ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 qTop 𝐹) ↾t 𝑈)
132131isopn2 21324 . . . . . . . . . 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 584 . . . . . . . . 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 258 . . . . . . . 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 953 . . . . . . 7 ((𝜑 ∧ (𝑥𝑈 ∧ (𝐹𝑥) ∈ (𝐽t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
138137expr 457 . . . . . 6 ((𝜑𝑥𝑈) → ((𝐹𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
13957, 138sylbid 241 . . . . 5 ((𝜑𝑥𝑈) → (((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140139expimpd 454 . . . 4 (𝜑 → ((𝑥𝑈 ∧ ((𝐹𝐴) “ 𝑥) ∈ (𝐽t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
14148, 140sylbid 241 . . 3 (𝜑 → (𝑥 ∈ ((𝐽t 𝐴) qTop (𝐹𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
142141ssrdv 3895 . 2 (𝜑 → ((𝐽t 𝐴) qTop (𝐹𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
14336, 142eqssd 3906 1 (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∪ cuni 4745  ◡ccnv 5442  dom cdm 5443  ran crn 5444   ↾ cres 5445   “ cima 5446  Fun wfun 6219   Fn wfn 6220  –onto→wfo 6223  ‘cfv 6225  (class class class)co 7016   ↾t crest 16523   qTop cqtop 16605  Topctop 21185  TopOnctopon 21202  Clsdccld 21308   Cn ccn 21516 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-fin 8361  df-fi 8721  df-rest 16525  df-topgen 16546  df-qtop 16609  df-top 21186  df-topon 21203  df-bases 21238  df-cld 21311  df-cn 21519 This theorem is referenced by: (None)
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