Theorem divsfvalg 13475

Description: Value of the function in qusval 13469. (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 6780	. 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 6788	. . 3 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
7	4, 6	ssexd 4234	. 2 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
8	1, 2, 3, 7	fvmptd3 5749	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ↦ cmpt 4155  ‘cfv 5333   Er wer 6742  [cec 6743

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fv 5341  df-er 6745  df-ec 6747

This theorem is referenced by:  ercpbllemg  13476  qusaddvallemg  13479  qusgrp2  13763  qusring2  14143