imastopn - Metamath Proof Explorer

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Theorem imastopn 23707

Description: The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)

**Hypotheses**
Ref	Expression
imastps.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imastps.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imastps.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imastopn.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
imastopn.j	⊢ 𝐽 = (TopOpen‘𝑅)
imastopn.o	⊢ 𝑂 = (TopOpen‘𝑈)

**Assertion**
Ref	Expression
imastopn	⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹))

Proof of Theorem imastopn Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imastps.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imastps.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imastps.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imastopn.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5		imastopn.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑅)
6		eqid 2741	. . . . . . 7 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈)
7	1, 2, 3, 4, 5, 6	imastset 17481	. . . . . 6 ⊢ (𝜑 → (TopSet‘𝑈) = (𝐽 qTop 𝐹))
8	5	fvexi 6845	. . . . . . 7 ⊢ 𝐽 ∈ V
9		fofn 6745	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
11		fvex 6844	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
12	2, 11	eqeltrdi 2849	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V)
13		fnex 7165	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑉 ∈ V) → 𝐹 ∈ V)
14	10, 12, 13	syl2anc 591	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
15		eqid 2741	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	qtopval 23682	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((^◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽})
17	8, 14, 16	sylancr 594	. . . . . 6 ⊢ (𝜑 → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((^◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽})
18	7, 17	eqtrd 2776	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑈) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((^◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽})
19		ssrab2 4014	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((^◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (𝐹 “ ∪ 𝐽)
20		imassrn 6030	. . . . . . . 8 ⊢ (𝐹 “ ∪ 𝐽) ⊆ ran 𝐹
21		forn 6746	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
22	3, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
23	1, 2, 3, 4	imasbas 17471	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
24	22, 23	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
25	20, 24	sseqtrid 3959	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈))
26	25	sspwd 4545	. . . . . 6 ⊢ (𝜑 → 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈))
27	19, 26	sstrid 3928	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((^◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (Base‘𝑈))
28	18, 27	eqsstrd 3951	. . . 4 ⊢ (𝜑 → (TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈))
29		eqid 2741	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
30	29, 6	topnid 17393	. . . 4 ⊢ ((TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈) → (TopSet‘𝑈) = (TopOpen‘𝑈))
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → (TopSet‘𝑈) = (TopOpen‘𝑈))
32		imastopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑈)
33	31, 32	eqtr4di 2794	. 2 ⊢ (𝜑 → (TopSet‘𝑈) = 𝑂)
34	33, 7	eqtr3d 2778	1 ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1548 ∈ wcel 2121 {crab 3393 Vcvv 3433 ∩ cin 3884 ⊆ wss 3885 𝒫 cpw 4532 ∪ cuni 4841 ^◡ccnv 5620 ran crn 5622 “ cima 5624 Fn wfn 6484 –onto→wfo 6487 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 TopSetcts 17221 TopOpenctopn 17379 qTop cqtop 17462 “_s cimas 17463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-rest 17380 df-topn 17381 df-qtop 17466 df-imas 17467

This theorem is referenced by: imastps 23708 xpstopnlem2 23798 qustgpopn 24107 qustgplem 24108 qustgphaus 24110 imasf1oxms 24476

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