Theorem qtopval2 21988

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–onto→𝑌 ∧ 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 1129	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		fof 6458	. . . . 5 ⊢ (𝐹:𝑍–onto→𝑌 → 𝐹:𝑍⟶𝑌)
3	2	3ad2ant2 1127	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍⟶𝑌)
4		qtopval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5		uniexg 7325	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
6	5	3ad2ant1 1126	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
7	4, 6	syl5eqel 2887	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V)
8		simp3 1131	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋)
9	7, 8	ssexd 5119	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V)
10		fex 6855	. . . 4 ⊢ ((𝐹:𝑍⟶𝑌 ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
11	3, 9, 10	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹 ∈ V)
12	4	qtopval 21987	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
13	1, 11, 12	syl2anc 584	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
14		imassrn 5817	. . . . . 6 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
15		forn 6461	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → ran 𝐹 = 𝑌)
16	15	3ad2ant2 1127	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ran 𝐹 = 𝑌)
17	14, 16	sseqtrid 3940	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) ⊆ 𝑌)
18		foima 6463	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → (𝐹 “ 𝑍) = 𝑌)
19	18	3ad2ant2 1127	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) = 𝑌)
20		imass2 5841	. . . . . . 7 ⊢ (𝑍 ⊆ 𝑋 → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
21	8, 20	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
22	19, 21	eqsstrrd 3927	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ (𝐹 “ 𝑋))
23	17, 22	eqssd 3906	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) = 𝑌)
24	23	pweqd 4458	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝒫 (𝐹 “ 𝑋) = 𝒫 𝑌)
25		cnvimass 5825	. . . . . . 7 ⊢ (◡𝐹 “ 𝑠) ⊆ dom 𝐹
26	25, 3	fssdm 6398	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑍)
27	26, 8	sstrd 3899	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑋)
28		df-ss 3874	. . . . 5 ⊢ ((◡𝐹 “ 𝑠) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
29	27, 28	sylib 219	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
30	29	eleq1d 2867	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (◡𝐹 “ 𝑠) ∈ 𝐽))
31	24, 30	rabeqbidv 3430	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})
32	13, 31	eqtrd 2831	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  {crab 3109  Vcvv 3437   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745  ^◡ccnv 5442  ran crn 5444   “ cima 5446  ⟶wf 6221  –onto→wfo 6223  (class class class)co 7016   qTop cqtop 16605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-qtop 16609

This theorem is referenced by:  elqtop  21989