Theorem divsfvalg 12915

Description: Value of the function in qusval 12909. (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 6624	. 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 6632	. . 3 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
7	4, 6	ssexd 4170	. 2 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
8	1, 2, 3, 7	fvmptd3 5652	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ↦ cmpt 4091  ‘cfv 5255   Er wer 6586  [cec 6587

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 4148  ax-pow 4204  ax-pr 4239

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 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fv 5263  df-er 6589  df-ec 6591

This theorem is referenced by:  ercpbllemg  12916  qusaddvallemg  12919  qusgrp2  13186  qusring2  13565