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Theorem fnmpo 8073
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 3490 . . 3 (𝐶𝑉𝐶 ∈ V)
212ralimi 3120 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶 ∈ V)
3 fmpo.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 8072 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹:(𝐴 × 𝐵)⟶V)
5 dffn2 6724 . . 3 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
64, 5bitr4i 278 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹 Fn (𝐴 × 𝐵))
72, 6sylib 217 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wral 3058  Vcvv 3471   × cxp 5676   Fn wfn 6543  wf 6544  cmpo 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994
This theorem is referenced by:  fnmpoi  8074  dmmpoga  8077  dmmpogaOLD  8078  fnmpoovd  8092  fsplitfpar  8123  genpdm  11026  isofn  17758  brric  20443  mpocti  32510  f1od2  32516  cnre2csqima  33512  aks6d1c6lem3  41644  elrnmpoid  44601  smflimlem3  46161  smflimlem6  46164
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