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Theorem fvmpti 6268
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 6267 . . 3 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
4 fvi 6242 . . . 4 (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
54adantl 482 . . 3 ((𝐴𝐷𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
63, 5eqtr4d 2657 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
71eleq1d 2684 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
82dmmpt 5618 . . . . . . . 8 dom 𝐹 = {𝑥𝐷𝐵 ∈ V}
97, 8elrab2 3360 . . . . . . 7 (𝐴 ∈ dom 𝐹 ↔ (𝐴𝐷𝐶 ∈ V))
109baib 943 . . . . . 6 (𝐴𝐷 → (𝐴 ∈ dom 𝐹𝐶 ∈ V))
1110notbid 308 . . . . 5 (𝐴𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12 ndmfv 6205 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
1311, 12syl6bir 244 . . . 4 (𝐴𝐷 → (¬ 𝐶 ∈ V → (𝐹𝐴) = ∅))
1413imp 445 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
15 fvprc 6172 . . . 4 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1615adantl 482 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
1714, 16eqtr4d 2657 . 2 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
186, 17pm2.61dan 831 1 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cmpt 4720   I cid 5013  dom cdm 5104  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884
This theorem is referenced by:  fvmpt2i  6277  fvmptex  6281  sumeq2ii  14404  summolem3  14426  fsumf1o  14435  isumshft  14552  prodeq2ii  14624  prodmolem3  14644  fprodf1o  14657
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