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Theorem fvmpti 6749
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 6748 . . 3 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
4 fvi 6722 . . . 4 (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
54adantl 485 . . 3 ((𝐴𝐷𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
63, 5eqtr4d 2860 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
71eleq1d 2898 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
82dmmpt 6072 . . . . . . . 8 dom 𝐹 = {𝑥𝐷𝐵 ∈ V}
97, 8elrab2 3658 . . . . . . 7 (𝐴 ∈ dom 𝐹 ↔ (𝐴𝐷𝐶 ∈ V))
109baib 539 . . . . . 6 (𝐴𝐷 → (𝐴 ∈ dom 𝐹𝐶 ∈ V))
1110notbid 321 . . . . 5 (𝐴𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12 ndmfv 6682 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
1311, 12syl6bir 257 . . . 4 (𝐴𝐷 → (¬ 𝐶 ∈ V → (𝐹𝐴) = ∅))
1413imp 410 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
15 fvprc 6645 . . . 4 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1615adantl 485 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
1714, 16eqtr4d 2860 . 2 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
186, 17pm2.61dan 812 1 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  cmpt 5122   I cid 5436  dom cdm 5532  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342
This theorem is referenced by:  fvmpt2i  6760  fvmptex  6764  sumeq2ii  15041  summolem3  15062  fsumf1o  15071  isumshft  15185  prodeq2ii  15258  prodmolem3  15278  fprodf1o  15291
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