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Theorem fvmpti 6079
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 6078 . . 3 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
4 fvi 6054 . . . 4 (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
54adantl 480 . . 3 ((𝐴𝐷𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
63, 5eqtr4d 2551 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
71eleq1d 2576 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
82dmmpt 5437 . . . . . . . 8 dom 𝐹 = {𝑥𝐷𝐵 ∈ V}
97, 8elrab2 3237 . . . . . . 7 (𝐴 ∈ dom 𝐹 ↔ (𝐴𝐷𝐶 ∈ V))
109baib 941 . . . . . 6 (𝐴𝐷 → (𝐴 ∈ dom 𝐹𝐶 ∈ V))
1110notbid 306 . . . . 5 (𝐴𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12 ndmfv 6017 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
1311, 12syl6bir 242 . . . 4 (𝐴𝐷 → (¬ 𝐶 ∈ V → (𝐹𝐴) = ∅))
1413imp 443 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
15 fvprc 5986 . . . 4 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1615adantl 480 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
1714, 16eqtr4d 2551 . 2 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
186, 17pm2.61dan 827 1 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  c0 3777  cmpt 4541   I cid 4842  dom cdm 4932  cfv 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5658  df-fun 5696  df-fv 5702
This theorem is referenced by:  fvmpt2i  6088  fvmptex  6092  sumeq2ii  14144  summolem3  14165  fsumf1o  14174  isumshft  14283  prodeq2ii  14355  prodmolem3  14375  fprodf1o  14388
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