Theorem fvmptd4 7010

Description: Deduction version of fvmpt 6986 (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 481	. 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 6993	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5201  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539

This theorem is referenced by:  selvval  22073  mhpmulcl  22087  psdval  22097  psdcoef  22098  evl1deg1  33589  evl1deg2  33590  evl1deg3  33591  2sqr3minply  33814  cos9thpiminply  33822  evlsvval  42583  evlsvvval  42586  evlsvarval  42588  selvvvval  42608  prjcrvval  42655  dvnprodlem1  45975