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Theorem fvmpti 6532
Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fvmptg.1 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptg.2 𝐹 = (𝑥𝐷𝐵)
Assertion
Ref Expression
fvmpti (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpti
StepHypRef Expression
1 fvmptg.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
2 fvmptg.2 . . . 4 𝐹 = (𝑥𝐷𝐵)
31, 2fvmptg 6531 . . 3 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
4 fvi 6506 . . . 4 (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
54adantl 475 . . 3 ((𝐴𝐷𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
63, 5eqtr4d 2864 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
71eleq1d 2891 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
82dmmpt 5875 . . . . . . . 8 dom 𝐹 = {𝑥𝐷𝐵 ∈ V}
97, 8elrab2 3589 . . . . . . 7 (𝐴 ∈ dom 𝐹 ↔ (𝐴𝐷𝐶 ∈ V))
109baib 531 . . . . . 6 (𝐴𝐷 → (𝐴 ∈ dom 𝐹𝐶 ∈ V))
1110notbid 310 . . . . 5 (𝐴𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12 ndmfv 6467 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
1311, 12syl6bir 246 . . . 4 (𝐴𝐷 → (¬ 𝐶 ∈ V → (𝐹𝐴) = ∅))
1413imp 397 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
15 fvprc 6430 . . . 4 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1615adantl 475 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
1714, 16eqtr4d 2864 . 2 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
186, 17pm2.61dan 847 1 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  c0 4146  cmpt 4954   I cid 5251  dom cdm 5346  cfv 6127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fv 6135
This theorem is referenced by:  fvmpt2i  6542  fvmptex  6546  sumeq2ii  14807  summolem3  14829  fsumf1o  14838  isumshft  14952  prodeq2ii  15023  prodmolem3  15043  fprodf1o  15056
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