Theorem qtopval2 23182

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 1137	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		fof 6802	. . . . 5 ⊢ (𝐹:𝑍–onto→𝑌 → 𝐹:𝑍⟶𝑌)
3	2	3ad2ant2 1135	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍⟶𝑌)
4		qtopval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5		uniexg 7725	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
6	5	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
7	4, 6	eqeltrid 2838	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V)
8		simp3 1139	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋)
9	7, 8	ssexd 5323	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V)
10	3, 9	fexd 7224	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹 ∈ V)
11	4	qtopval 23181	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
12	1, 10, 11	syl2anc 585	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
13		imassrn 6068	. . . . . 6 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
14		forn 6805	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → ran 𝐹 = 𝑌)
15	14	3ad2ant2 1135	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ran 𝐹 = 𝑌)
16	13, 15	sseqtrid 4033	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) ⊆ 𝑌)
17		foima 6807	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → (𝐹 “ 𝑍) = 𝑌)
18	17	3ad2ant2 1135	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) = 𝑌)
19		imass2 6098	. . . . . . 7 ⊢ (𝑍 ⊆ 𝑋 → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
20	8, 19	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
21	18, 20	eqsstrrd 4020	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ (𝐹 “ 𝑋))
22	16, 21	eqssd 3998	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) = 𝑌)
23	22	pweqd 4618	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝒫 (𝐹 “ 𝑋) = 𝒫 𝑌)
24		cnvimass 6077	. . . . . . 7 ⊢ (◡𝐹 “ 𝑠) ⊆ dom 𝐹
25	24, 3	fssdm 6734	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑍)
26	25, 8	sstrd 3991	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑋)
27		df-ss 3964	. . . . 5 ⊢ ((◡𝐹 “ 𝑠) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
28	26, 27	sylib 217	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
29	28	eleq1d 2819	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (◡𝐹 “ 𝑠) ∈ 𝐽))
30	23, 29	rabeqbidv 3450	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})
31	12, 30	eqtrd 2773	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  ^◡ccnv 5674  ran crn 5676   “ cima 5678  ⟶wf 6536  –onto→wfo 6538  (class class class)co 7404   qTop cqtop 17445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-qtop 17449

This theorem is referenced by:  elqtop  23183