Theorem fvmptdf 7021

Description: Deduction version of fvmptd 7022 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 6908	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		fvmptd.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
4		fvmptdf.p	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		nfcsb1v 3932	. . . . . 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 3921	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
10	9	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		fvmptdf.a	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12	11	nfeq2 2920	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝐴
13	4, 12	nfan 1896	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐴)
14	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → Ⅎ𝑥𝐶)
15		vex 3481	. . . . . . 7 ⊢ 𝑦 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 ∈ V)
17		eqtr 2757	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝐴) → 𝑥 = 𝐴)
18	17	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴)
19		fvmptd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
20	18, 19	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → 𝐵 = 𝐶)
21	20	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
22	13, 14, 16, 21	csbiedf 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
23	4, 6, 8, 3, 10, 22	csbie2df 4448	. . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
24		fvmptd.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
25	23, 24	eqeltrd 2838	. . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉)
26		eqid 2734	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
27	26	fvmpts 7018	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
28	3, 25, 27	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
29	2, 28, 23	3eqtrd 2778	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536  Ⅎwnf 1779   ∈ wcel 2105  Ⅎwnfc 2887  Vcvv 3477  ⦋csb 3907   ↦ cmpt 5230  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570

This theorem is referenced by:  fvmptd  7022  symgval  19402  cfsetsnfsetf  47007  1arymaptfo  48492  2arymaptfo  48503