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Theorem fvmpti 6239
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 6238 . . 3 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
4 fvi 6213 . . . 4 (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
54adantl 482 . . 3 ((𝐴𝐷𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
63, 5eqtr4d 2663 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = ( I ‘𝐶))
71eleq1d 2688 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
82dmmpt 5592 . . . . . . . 8 dom 𝐹 = {𝑥𝐷𝐵 ∈ V}
97, 8elrab2 3353 . . . . . . 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 6176 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
1311, 12syl6bir 244 . . . 4 (𝐴𝐷 → (¬ 𝐶 ∈ V → (𝐹𝐴) = ∅))
1413imp 445 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
15 fvprc 6144 . . . 4 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1615adantl 482 . . 3 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
1714, 16eqtr4d 2663 . 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 1480  wcel 1992  Vcvv 3191  c0 3896  cmpt 4678   I cid 4989  dom cdm 5079  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858
This theorem is referenced by:  fvmpt2i  6248  fvmptex  6252  sumeq2ii  14352  summolem3  14373  fsumf1o  14382  isumshft  14491  prodeq2ii  14563  prodmolem3  14583  fprodf1o  14596
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