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Theorem fnmpo 7753
Description: Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
Hypothesis
Ref Expression
fmpo.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
fnmpo (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem fnmpo
StepHypRef Expression
1 elex 3462 . . 3 (𝐶𝑉𝐶 ∈ V)
212ralimi 3132 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶 ∈ V)
3 fmpo.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 7752 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹:(𝐴 × 𝐵)⟶V)
5 dffn2 6493 . . 3 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
64, 5bitr4i 281 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹 Fn (𝐴 × 𝐵))
72, 6sylib 221 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444   × cxp 5521   Fn wfn 6323  wf 6324  cmpo 7141
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676
This theorem is referenced by:  fnmpoi  7754  dmmpoga  7757  dmmpogaOLD  7758  fnmpoovd  7769  fsplitfpar  7801  genpdm  10417  isofn  17040  brric  19495  mpocti  30480  f1od2  30486  cnre2csqima  31262  elrnmpoid  41847  smflimlem3  43393  smflimlem6  43396
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