Theorem divsfvalg 13402

Description: Value of the function in qusval 13396. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
ercpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
ercpbl.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
divsfvalg	⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Distinct variable groups:   𝑥, ∼   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem divsfvalg

Step	Hyp	Ref	Expression
1		ercpbl.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
2		eceq1 6732	. 2 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
3		ercpbl.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ercpbl.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
5		ercpbl.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
6	5	ecss 6740	. . 3 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
7	4, 6	ssexd 4227	. 2 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
8	1, 2, 3, 7	fvmptd3 5736	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ↦ cmpt 4148  ‘cfv 5324   Er wer 6694  [cec 6695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332  df-er 6697  df-ec 6699

This theorem is referenced by:  ercpbllemg  13403  qusaddvallemg  13406  qusgrp2  13690  qusring2  14069