Theorem fvmptd4 40210

Description: Deduction version of fvmpt 6875 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)

Hypotheses

Ref	Expression
fvmptd4.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvmptd4.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
fvmptd4.3	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fvmptd4.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fvmptd4	⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd4

Step	Hyp	Ref	Expression
1		fvmptd4.2	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
2		fvmptd4.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
3	2	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
4		fvmptd4.3	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5		fvmptd4.4	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
6	1, 3, 4, 5	fvmptd 6882	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  evlsvarval  40274  evlsbagval  40275  prjcrvval  40469