Theorem fvmptd4 7018

Description: Deduction version of fvmpt 6993 (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 487	. 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 7001	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548

This theorem is used by:  evlsvval  22270  evlsvvval  22273  selvval  22300  evlsvarval  22307  selvvvval  22322  mhpmulcl  22341  psdval  22351  psdcoef  22352  ishlg2  28900  evl1deg1  33889  evl1deg2  33890  evl1deg3  33891  selvply1rhmlema  33931  selvply1rhmlemb  33932  selvply1rhmlem3  33935  extvfval  33945  extvfv  33946  extvfvv  33947  evlvarval  33954  mplvrpmrhm  33960  esplyfval  33976  esplyind  33988  vieta  33993  extdgfialglem2  34106  2sqr3minply  34193  cos9thpiminply  34201  prjcrvval  43397  dvnprodlem1  46693