Theorem qtopval2 24070

Description: Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

Ref	Expression
qtopval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
qtopval2	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝑉,𝑠   𝑌,𝑠   𝑍,𝑠   𝑋,𝑠

Proof of Theorem qtopval2

Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		fof 6826	. . . . 5 ⊢ (𝐹:𝑍⟶onto𝑌 → 𝐹:𝑍⟶𝑌)
3	2	3ad2ant2 1152	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍⟶𝑌)
4		qtopval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5		uniexg 7789	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
6	5	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
7	4, 6	eqeltrid 2868	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V)
8		simp3 1156	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋)
9	7, 8	ssexd 5293	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V)
10	3, 9	fexd 7264	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹 ∈ V)
11	4	qtopval 24069	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
12	1, 10, 11	syl2anc 596	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
13		imassrn 6216	. . . . . 6 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
14		forn 6829	. . . . . . 7 ⊢ (𝐹:𝑍⟶onto𝑌 → ran 𝐹 = 𝑌)
15	14	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → ran 𝐹 = 𝑌)
16	13, 15	sseqtrid 3955	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) ⊆ 𝑌)
17		foima 6831	. . . . . . 7 ⊢ (𝐹:𝑍⟶onto𝑌 → (𝐹 “ 𝑍) = 𝑌)
18	17	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) = 𝑌)
19		imass2 6073	. . . . . . 7 ⊢ (𝑍 ⊆ 𝑋 → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
20	8, 19	syl 18	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
21	18, 20	eqsstrrd 3948	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ (𝐹 “ 𝑋))
22	16, 21	eqssd 3930	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) = 𝑌)
23	22	pweqd 4577	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝒫 (𝐹 “ 𝑋) = 𝒫 𝑌)
24		cnvimass 6217	. . . . . . 7 ⊢ (◡𝐹 “ 𝑠) ⊆ dom 𝐹
25	24, 3	fssdm 6759	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑍)
26	25, 8	sstrd 3923	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑋)
27		dfss2 4097	. . . . 5 ⊢ ((◡𝐹 “ 𝑠) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
28	26, 27	sylib 221	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
29	28	eleq1d 2849	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (◡𝐹 “ 𝑠) ∈ 𝐽))
30	23, 29	rabeqbidv 3433	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})
31	12, 30	eqtrd 2798	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍⟶onto𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4872  ^◡ccnv 5657  ran crn 5659   “ cima 5661  ⟶wf 6562  ⟶ontowfo 6564  (class class class)co 7451   qTop cqtop 17725

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-qtop 17729

This theorem is used by:  elqtop  24071