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Theorem fmpti 5321
 Description: Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
fmpt.1 𝐹 = (𝑥𝐴𝐶)
fmpti.2 (𝑥𝐴𝐶𝐵)
Assertion
Ref Expression
fmpti 𝐹:𝐴𝐵
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fmpti
StepHypRef Expression
1 fmpti.2 . . 3 (𝑥𝐴𝐶𝐵)
21rgen 2374 . 2 𝑥𝐴 𝐶𝐵
3 fmpt.1 . . 3 𝐹 = (𝑥𝐴𝐶)
43fmpt 5319 . 2 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
52, 4mpbi 133 1 𝐹:𝐴𝐵
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ↦ cmpt 3818  ⟶wf 4898 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910 This theorem is referenced by:  cjf  9447  ref  9455  imf  9456  absf  9706
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