Theorem fvmptd4 6961

Description: Deduction version of fvmpt 6937 (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 6944	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ↦ cmpt 5176  ‘cfv 6488

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496

This theorem is referenced by:  selvval  22053  mhpmulcl  22067  psdval  22077  psdcoef  22078  evl1deg1  33548  evl1deg2  33549  evl1deg3  33550  extvfval  33585  extvfv  33586  extvfvv  33587  mplvrpmrhm  33597  esplyfval  33606  esplyind  33615  extdgfialglem2  33729  2sqr3minply  33816  cos9thpiminply  33824  evlsvval  42681  evlsvvval  42684  evlsvarval  42686  selvvvval  42706  prjcrvval  42753  dvnprodlem1  46071