Theorem kqfvima 23672

Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7177, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqfvima	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqfvima

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23667	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		toponss 22869	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑋)
5	4	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
6		fnfvima 7177	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑈) → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈))
7	6	3expia 1121	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
8	3, 5, 7	syl2anc 584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
9		fnfun 6590	. . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
10		fvelima 6897	. . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴))
11	10	ex 412	. . . 4 ⊢ (Fun 𝐹 → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴)))
12	3, 9, 11	3syl 18	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴)))
13		simpl1 1192	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
14	5	sselda 3931	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑋)
15		simpl3 1194	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐴 ∈ 𝑋)
16	1	kqfeq 23666	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
17	13, 14, 15, 16	syl3anc 1373	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
18		eleq2 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
19		eleq2 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
20	18, 19	bibi12d 345	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤)))
21	20	cbvralvw 3212	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤))
22	17, 21	bitrdi 287	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤)))
23		simpl2 1193	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝐽)
24		eleq2 2823	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑈))
25		eleq2 2823	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑈))
26	24, 25	bibi12d 345	. . . . . . . 8 ⊢ (𝑤 = 𝑈 → ((𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) ↔ (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
27	26	rspcv 3570	. . . . . . 7 ⊢ (𝑈 ∈ 𝐽 → (∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
28	23, 27	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → (∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
29	22, 28	sylbid 240	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
30		simpr 484	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
31		biimp 215	. . . . 5 ⊢ ((𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈) → (𝑧 ∈ 𝑈 → 𝐴 ∈ 𝑈))
32	29, 30, 31	syl6ci 71	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) → 𝐴 ∈ 𝑈))
33	32	rexlimdva 3135	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴) → 𝐴 ∈ 𝑈))
34	12, 33	syld 47	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → 𝐴 ∈ 𝑈))
35	8, 34	impbid 212	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  {crab 3397   ⊆ wss 3899   ↦ cmpt 5177   “ cima 5625  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490  TopOnctopon 22852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-topon 22853

This theorem is referenced by:  kqsat  23673  kqdisj  23674  kqcldsat  23675  kqt0lem  23678  isr0  23679  regr1lem  23681  kqreglem1  23683  kqreglem2  23684