Theorem qtopval2 22847

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 1135	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		fof 6688	. . . . 5 ⊢ (𝐹:𝑍–onto→𝑌 → 𝐹:𝑍⟶𝑌)
3	2	3ad2ant2 1133	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍⟶𝑌)
4		qtopval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5		uniexg 7593	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
6	5	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
7	4, 6	eqeltrid 2843	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V)
8		simp3 1137	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋)
9	7, 8	ssexd 5248	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V)
10	3, 9	fexd 7103	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹 ∈ V)
11	4	qtopval 22846	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
12	1, 10, 11	syl2anc 584	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
13		imassrn 5980	. . . . . 6 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
14		forn 6691	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → ran 𝐹 = 𝑌)
15	14	3ad2ant2 1133	. . . . . 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 6693	. . . . . . 7 ⊢ (𝐹:𝑍–onto→𝑌 → (𝐹 “ 𝑍) = 𝑌)
18	17	3ad2ant2 1133	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑍) = 𝑌)
19		imass2 6010	. . . . . . 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 3960	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ (𝐹 “ 𝑋))
22	16, 21	eqssd 3938	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐹 “ 𝑋) = 𝑌)
23	22	pweqd 4552	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝒫 (𝐹 “ 𝑋) = 𝒫 𝑌)
24		cnvimass 5989	. . . . . . 7 ⊢ (◡𝐹 “ 𝑠) ⊆ dom 𝐹
25	24, 3	fssdm 6620	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑍)
26	25, 8	sstrd 3931	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (◡𝐹 “ 𝑠) ⊆ 𝑋)
27		df-ss 3904	. . . . 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 2823	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (◡𝐹 “ 𝑠) ∈ 𝐽))
30	23, 29	rabeqbidv 3420	. 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 2778	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  ^◡ccnv 5588  ran crn 5590   “ cima 5592  ⟶wf 6429  –onto→wfo 6431  (class class class)co 7275   qTop cqtop 17214

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-qtop 17218

This theorem is referenced by:  elqtop  22848