Theorem divsfvalg 12912

Description: Value of the function in qusval 12906. (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 6622	. 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 6630	. . 3 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
7	4, 6	ssexd 4169	. 2 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
8	1, 2, 3, 7	fvmptd3 5651	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ↦ cmpt 4090  ‘cfv 5254   Er wer 6584  [cec 6585

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262  df-er 6587  df-ec 6589

This theorem is referenced by:  ercpbllemg  12913  qusaddvallemg  12916  qusgrp2  13183  qusring2  13562