Theorem fvmptf 5206

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5191 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
fvmptf.1	⊢ ℲxA
fvmptf.2	⊢ Ⅎx𝐶
fvmptf.3	⊢ (x = A → B = 𝐶)
fvmptf.4	⊢ 𝐹 = (x ∈ 𝐷 ↦ B)

Assertion

Ref	Expression
fvmptf	⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘A) = 𝐶)

Distinct variable group:   x,𝐷

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

Proof of Theorem fvmptf

Step	Hyp	Ref	Expression
1		elex 2560	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
2		fvmptf.1	. . . 4 ⊢ ℲxA
3		fvmptf.2	. . . . . 6 ⊢ Ⅎx𝐶
4	3	nfel1 2185	. . . . 5 ⊢ Ⅎx 𝐶 ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ 𝐹 = (x ∈ 𝐷 ↦ B)
6		nfmpt1 3841	. . . . . . . 8 ⊢ Ⅎx(x ∈ 𝐷 ↦ B)
7	5, 6	nfcxfr 2172	. . . . . . 7 ⊢ Ⅎx𝐹
8	7, 2	nffv 5128	. . . . . 6 ⊢ Ⅎx(𝐹‘A)
9	8, 3	nfeq 2182	. . . . 5 ⊢ Ⅎx(𝐹‘A) = 𝐶
10	4, 9	nfim 1461	. . . 4 ⊢ Ⅎx(𝐶 ∈ V → (𝐹‘A) = 𝐶)
11		fvmptf.3	. . . . . 6 ⊢ (x = A → B = 𝐶)
12	11	eleq1d 2103	. . . . 5 ⊢ (x = A → (B ∈ V ↔ 𝐶 ∈ V))
13		fveq2 5121	. . . . . 6 ⊢ (x = A → (𝐹‘x) = (𝐹‘A))
14	13, 11	eqeq12d 2051	. . . . 5 ⊢ (x = A → ((𝐹‘x) = B ↔ (𝐹‘A) = 𝐶))
15	12, 14	imbi12d 223	. . . 4 ⊢ (x = A → ((B ∈ V → (𝐹‘x) = B) ↔ (𝐶 ∈ V → (𝐹‘A) = 𝐶)))
16	5	fvmpt2 5197	. . . . 5 ⊢ ((x ∈ 𝐷 ∧ B ∈ V) → (𝐹‘x) = B)
17	16	ex 108	. . . 4 ⊢ (x ∈ 𝐷 → (B ∈ V → (𝐹‘x) = B))
18	2, 10, 15, 17	vtoclgaf 2612	. . 3 ⊢ (A ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘A) = 𝐶))
19	1, 18	syl5 28	. 2 ⊢ (A ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘A) = 𝐶))
20	19	imp 115	1 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘A) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Ⅎwnfc 2162  Vcvv 2551   ↦ cmpt 3809  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853

This theorem is referenced by: (None)