Theorem qtopval2 24008

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 1154	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		fof 6794	. . . . 5 ⊢ (𝐹:𝑍–onto→𝑌 → 𝐹:𝑍⟶𝑌)
3	2	3ad2ant2 1152	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍⟶𝑌)
4		qtopval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5		uniexg 7755	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
6	5	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
7	4, 6	eqeltrid 2865	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V)
8		simp3 1156	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋)
9	7, 8	ssexd 5286	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V)
10	3, 9	fexd 7231	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹 ∈ V)
11	4	qtopval 24007	. . 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–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
13		imassrn 6196	. . . . . 6 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
14		forn 6797	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → ran 𝐹 = 𝑌)
15	14	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ran 𝐹 = 𝑌)
16	13, 15	sseqtrid 3973	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) ⊆ 𝑌)
17		foima 6799	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → (𝐹 “ 𝑍) = 𝑌)
18	17	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) = 𝑌)
19		imass2 6055	. . . . . . 7 ⊢ (𝑍 ⊆ 𝑋 → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
20	8, 19	syl 18	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) ⊆ (𝐹 “ 𝑋))
21	18, 20	eqsstrrd 3966	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ (𝐹 “ 𝑋))
22	16, 21	eqssd 3948	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) = 𝑌)
23	22	pweqd 4574	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝒫 (𝐹 “ 𝑋) = 𝒫 𝑌)
24		cnvimass 6197	. . . . . . 7 ⊢ (◡𝐹 “ 𝑠) ⊆ dom 𝐹
25	24, 3	fssdm 6727	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑍)
26	25, 8	sstrd 3941	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑋)
27		dfss2 3917	. . . . 5 ⊢ ((◡𝐹 “ 𝑠) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
28	26, 27	sylib 221	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ((◡𝐹 “ 𝑠) ∩ 𝑋) = (◡𝐹 “ 𝑠))
29	28	eleq1d 2846	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (◡𝐹 “ 𝑠) ∈ 𝐽))
30	23, 29	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})
31	12, 30	eqtrd 2796	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ^◡ccnv 5650  ran crn 5652   “ cima 5654  ⟶wf 6533  –onto→wfo 6535  (class class class)co 7418   qTop cqtop 17668

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-qtop 17672

This theorem is used by:  elqtop  24009