Theorem fvmpt2df 45701

Description: Deduction version of fvmpt2 6959. (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 6841	. 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 6948	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
7	3, 4, 6	syl2an2 687	. 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 1542   ∈ wcel 2114  Ⅎwnfc 2883   ↦ cmpt 5166  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506

This theorem is referenced by:  fsupdm  47270