Theorem fvmpt2df 45512

Description: Deduction version of fvmpt2 6952. (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 6835	. 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 6942	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
7	3, 4, 6	syl2an2 686	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
8	2, 7	eqtrid 2783	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883   ↦ cmpt 5179  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  fsupdm  47082