Theorem fvmptdf 7004

Description: Deduction version of fvmptd 7005 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 6893	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		fvmptd.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
4		fvmptdf.p	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		nfcsb1v 3918	. . . . . 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 3907	. . . . . 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 2919	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝐴
13	4, 12	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐴)
14	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → Ⅎ𝑥𝐶)
15		vex 3477	. . . . . . 7 ⊢ 𝑦 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 ∈ V)
17		eqtr 2754	. . . . . . . . 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 592	. . . . . . 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 3924	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
23	4, 6, 8, 3, 10, 22	csbie2df 4440	. . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
24		fvmptd.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
25	23, 24	eqeltrd 2832	. . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉)
26		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
27	26	fvmpts 7001	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
28	3, 25, 27	syl2anc 583	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
29	2, 28, 23	3eqtrd 2775	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  Ⅎwnfc 2882  Vcvv 3473  ⦋csb 3893   ↦ cmpt 5231  ‘cfv 6543

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by:  fvmptd  7005  symgval  19278  cfsetsnfsetf  46067  1arymaptfo  47417  2arymaptfo  47428