Theorem fvmptd4 7014

Description: Deduction version of fvmpt 6989 (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 486	. 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 6997	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143   ↦ cmpt 5192  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is used by:  evlsvval  22250  evlsvvval  22253  selvval  22280  evlsvarval  22287  selvvvval  22302  mhpmulcl  22321  psdval  22331  psdcoef  22332  ishlg2  28880  evl1deg1  33875  evl1deg2  33876  evl1deg3  33877  selvply1rhmlema  33917  selvply1rhmlemb  33918  selvply1rhmlem3  33921  extvfval  33931  extvfv  33932  extvfvv  33933  evlvarval  33940  mplvrpmrhm  33946  esplyfval  33962  esplyind  33974  vieta  33979  extdgfialglem2  34092  2sqr3minply  34179  cos9thpiminply  34187  prjcrvval  43392  dvnprodlem1  46688