Theorem kqfvima 22862

Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7103, 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 22857	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1131	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		toponss 22057	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑋)
5	4	3adant3 1130	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
6		fnfvima 7103	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑈) → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈))
7	6	3expia 1119	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
8	3, 5, 7	syl2anc 583	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
9		fnfun 6529	. . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
10		fvelima 6829	. . . . 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 1189	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
14	5	sselda 3925	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑋)
15		simpl3 1191	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐴 ∈ 𝑋)
16	1	kqfeq 22856	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
17	13, 14, 15, 16	syl3anc 1369	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
18		eleq2 2828	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
19		eleq2 2828	. . . . . . . . 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 3380	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤))
22	17, 21	bitrdi 286	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤)))
23		simpl2 1190	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝐽)
24		eleq2 2828	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑈))
25		eleq2 2828	. . . . . . . . 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 3555	. . . . . . 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 239	. . . . 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 214	. . . . 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 3214	. . 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 211	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066  {crab 3069   ⊆ wss 3891   ↦ cmpt 5161   “ cima 5591  Fun wfun 6424   Fn wfn 6425  ‘cfv 6430  TopOnctopon 22040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-topon 22041

This theorem is referenced by:  kqsat  22863  kqdisj  22864  kqcldsat  22865  kqt0lem  22868  isr0  22869  regr1lem  22871  kqreglem1  22873  kqreglem2  22874