Theorem fvmpti 5699

Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fvmptg.1	⊢ (x = A → B = C)
fvmptg.2	⊢ F = (x ∈ D ↦ B)

Assertion

Ref	Expression
fvmpti	⊢ (A ∈ D → (F ‘A) = ( I ‘C))

Distinct variable groups:   x,A   x,C   x,D

Allowed substitution hints:   B(x)   F(x)

Proof of Theorem fvmpti

Step	Hyp	Ref	Expression
1		fvmptg.1	. . . 4 ⊢ (x = A → B = C)
2		fvmptg.2	. . . 4 ⊢ F = (x ∈ D ↦ B)
3	1, 2	fvmptg 5698	. . 3 ⊢ ((A ∈ D ∧ C ∈ V) → (F ‘A) = C)
4		fvi 5442	. . . 4 ⊢ (C ∈ V → ( I ‘C) = C)
5	4	adantl 452	. . 3 ⊢ ((A ∈ D ∧ C ∈ V) → ( I ‘C) = C)
6	3, 5	eqtr4d 2388	. 2 ⊢ ((A ∈ D ∧ C ∈ V) → (F ‘A) = ( I ‘C))
7	1	eleq1d 2419	. . . . . . . 8 ⊢ (x = A → (B ∈ V ↔ C ∈ V))
8	2	dmmpt 5683	. . . . . . . 8 ⊢ dom F = {x ∈ D ∣ B ∈ V}
9	7, 8	elrab2 2996	. . . . . . 7 ⊢ (A ∈ dom F ↔ (A ∈ D ∧ C ∈ V))
10	9	baib 871	. . . . . 6 ⊢ (A ∈ D → (A ∈ dom F ↔ C ∈ V))
11	10	notbid 285	. . . . 5 ⊢ (A ∈ D → (¬ A ∈ dom F ↔ ¬ C ∈ V))
12		ndmfv 5349	. . . . 5 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
13	11, 12	syl6bir 220	. . . 4 ⊢ (A ∈ D → (¬ C ∈ V → (F ‘A) = ∅))
14	13	imp 418	. . 3 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ∅)
15		fvprc 5325	. . . 4 ⊢ (¬ C ∈ V → ( I ‘C) = ∅)
16	15	adantl 452	. . 3 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → ( I ‘C) = ∅)
17	14, 16	eqtr4d 2388	. 2 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ( I ‘C))
18	6, 17	pm2.61dan 766	1 ⊢ (A ∈ D → (F ‘A) = ( I ‘C))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∅c0 3550   I cid 4763  dom cdm 4772   ‘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-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:  fvmpt2i  5703  fvmptex  5721