Theorem fvmptf 5722

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5698 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 2867	. . 3 ⊢ (C ∈ V → C ∈ V)
2		fvmptf.1	. . . 4 ⊢ ℲxA
3		fvmptf.2	. . . . . 6 ⊢ ℲxC
4	3	nfel1 2499	. . . . 5 ⊢ Ⅎx C ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ F = (x ∈ D ↦ B)
6		nfmpt1 5672	. . . . . . . 8 ⊢ Ⅎx(x ∈ D ↦ B)
7	5, 6	nfcxfr 2486	. . . . . . 7 ⊢ ℲxF
8	7, 2	nffv 5334	. . . . . 6 ⊢ Ⅎx(F ‘A)
9	8, 3	nfeq 2496	. . . . 5 ⊢ Ⅎx(F ‘A) = C
10	4, 9	nfim 1813	. . . 4 ⊢ Ⅎx(C ∈ V → (F ‘A) = C)
11		fvmptf.3	. . . . . 6 ⊢ (x = A → B = C)
12	11	eleq1d 2419	. . . . 5 ⊢ (x = A → (B ∈ V ↔ C ∈ V))
13		fveq2 5328	. . . . . 6 ⊢ (x = A → (F ‘x) = (F ‘A))
14	13, 11	eqeq12d 2367	. . . . 5 ⊢ (x = A → ((F ‘x) = B ↔ (F ‘A) = C))
15	12, 14	imbi12d 311	. . . 4 ⊢ (x = A → ((B ∈ V → (F ‘x) = B) ↔ (C ∈ V → (F ‘A) = C)))
16	5	fvmpt2 5704	. . . . 5 ⊢ ((x ∈ D ∧ B ∈ V) → (F ‘x) = B)
17	16	ex 423	. . . 4 ⊢ (x ∈ D → (B ∈ V → (F ‘x) = B))
18	2, 10, 15, 17	vtoclgaf 2919	. . 3 ⊢ (A ∈ D → (C ∈ V → (F ‘A) = C))
19	1, 18	syl5 28	. 2 ⊢ (A ∈ D → (C ∈ V → (F ‘A) = C))
20	19	imp 418	1 ⊢ ((A ∈ D ∧ C ∈ V) → (F ‘A) = C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859   ‘cfv 4781   ↦ cmpt 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652

This theorem is referenced by:  fvmptnf  5723