Theorem kqfvima 23957

Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7235, 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 23952	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		toponss 23153	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑋)
5	4	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
6		fnfvima 7235	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑈) → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈))
7	6	3expia 1139	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑈 ⊆ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
8	3, 5, 7	syl2anc 596	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 → (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
9		fnfun 6636	. . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
10		fvelima 6947	. . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴))
11	10	ex 418	. . . 4 ⊢ (Fun 𝐹 → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴)))
12	3, 9, 11	3syl 19	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → ∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴)))
13		simpl1 1210	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
14	5	sselda 3934	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑋)
15		simpl3 1212	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝐴 ∈ 𝑋)
16	1	kqfeq 23951	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
17	13, 14, 15, 16	syl3anc 1398	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
18		eleq2 2851	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
19		eleq2 2851	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
20	18, 19	bibi12d 348	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤)))
21	20	cbvralvw 3242	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤))
22	17, 21	bitrdi 290	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤)))
23		simpl2 1211	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝐽)
24		eleq2 2851	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑈))
25		eleq2 2851	. . . . . . . . 9 ⊢ (𝑤 = 𝑈 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑈))
26	24, 25	bibi12d 348	. . . . . . . 8 ⊢ (𝑤 = 𝑈 → ((𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) ↔ (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
27	26	rspcv 3575	. . . . . . 7 ⊢ (𝑈 ∈ 𝐽 → (∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
28	23, 27	syl 18	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → (∀𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ↔ 𝐴 ∈ 𝑤) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
29	22, 28	sylbid 243	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) → (𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)))
30		simpr 490	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
31		biimp 218	. . . . 5 ⊢ ((𝑧 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈) → (𝑧 ∈ 𝑈 → 𝐴 ∈ 𝑈))
32	29, 30, 31	syl6ci 72	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑈) → ((𝐹‘𝑧) = (𝐹‘𝐴) → 𝐴 ∈ 𝑈))
33	32	rexlimdva 3165	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (∃𝑧 ∈ 𝑈 (𝐹‘𝑧) = (𝐹‘𝐴) → 𝐴 ∈ 𝑈))
34	12, 33	syld 48	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ (𝐹 “ 𝑈) → 𝐴 ∈ 𝑈))
35	8, 34	impbid 215	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  {crab 3414   ⊆ wss 3902   ↦ cmpt 5190   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  TopOnctopon 23136

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-topon 23137

This theorem is used by:  kqsat  23958  kqdisj  23959  kqcldsat  23960  kqt0lem  23963  isr0  23964  regr1lem  23966  kqreglem1  23968  kqreglem2  23969