Theorem kqfvima 23617

Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7207, 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 23612	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		toponss 22814	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑋)
5	4	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
6		fnfvima 7207	. . . 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 6618	. . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
10		fvelima 6926	. . . . 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 3946	. . . . . . . 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 23611	. . . . . . . 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 2817	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
19		eleq2 2817	. . . . . . . . 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 3215	. . . . . . 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 2817	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑈))
25		eleq2 2817	. . . . . . . . 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 3584	. . . . . . 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 3134	. . 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405   ⊆ wss 3914   ↦ cmpt 5188   “ cima 5641  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  TopOnctopon 22797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-topon 22798

This theorem is referenced by:  kqsat  23618  kqdisj  23619  kqcldsat  23620  kqt0lem  23623  isr0  23624  regr1lem  23626  kqreglem1  23628  kqreglem2  23629