Theorem fvmpt2df 44549

Description: Deduction version of fvmpt2 7003. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypotheses

Ref	Expression
fvmpt2df.1	⊢ Ⅎ𝑥𝐴
fvmpt2df.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
fvmpt2df.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
fvmpt2df	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Proof of Theorem fvmpt2df

Step	Hyp	Ref	Expression
1		fvmpt2df.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fveq1i 6886	. 2 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
3		id 22	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
4		fvmpt2df.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
5		fvmpt2df.1	. . . 4 ⊢ Ⅎ𝑥𝐴
6	5	fvmpt2f 6993	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
7	3, 4, 6	syl2an2 683	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
8	2, 7	eqtrid 2778	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877   ↦ cmpt 5224  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545

This theorem is referenced by:  fsupdm  46130