Theorem fvmptdf 7001

Description: Deduction version of fvmptd 7002 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)

Hypotheses

Ref	Expression
fvmptd.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
fvmptd.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
fvmptd.3	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fvmptd.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)
fvmptdf.p	⊢ Ⅎ𝑥𝜑
fvmptdf.a	⊢ Ⅎ𝑥𝐴
fvmptdf.c	⊢ Ⅎ𝑥𝐶

Assertion

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

Distinct variable group:   𝑥,𝐷

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptdf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvmptd.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
2	1	fveq1d 6888	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		fvmptd.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
4		fvmptdf.p	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		nfcsb1v 3870	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵)
7		fvmptdf.c	. . . . . 6 ⊢ Ⅎ𝑥𝐶
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐶)
9		csbeq1a 3860	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
10	9	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		fvmptdf.a	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12	11	nfeq2 2939	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝐴
13	4, 12	nfan 1931	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐴)
14	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → Ⅎ𝑥𝐶)
15		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 ∈ V)
17		eqtr 2780	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝐴) → 𝑥 = 𝐴)
18	17	ancoms 464	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴)
19		fvmptd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
20	18, 19	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → 𝐵 = 𝐶)
21	20	anassrs 473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
22	13, 14, 16, 21	csbiedf 3876	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
23	4, 6, 8, 3, 10, 22	csbie2df 4400	. . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
24		fvmptd.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
25	23, 24	eqeltrd 2860	. . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉)
26		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
27	26	fvmpts 6998	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
28	3, 25, 27	syl2anc 596	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
29	2, 28, 23	3eqtrd 2799	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  Vcvv 3450  ⦋csb 3846   ↦ cmpt 5185  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547

This theorem is used by:  fvmptd  7002  symgval  19591  mplvrpmga  34224  cfsetsnfsetf  48222  1arymaptfo  49849  2arymaptfo  49860