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Theorem fnmpo 8002
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 3464 . . 3 (𝐶𝑉𝐶 ∈ V)
212ralimi 3127 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶 ∈ V)
3 fmpo.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 8001 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹:(𝐴 × 𝐵)⟶V)
5 dffn2 6671 . . 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 1542  wcel 2107  wral 3065  Vcvv 3446   × cxp 5632   Fn wfn 6492  wf 6493  cmpo 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923
This theorem is referenced by:  fnmpoi  8003  dmmpoga  8006  dmmpogaOLD  8007  fnmpoovd  8020  fsplitfpar  8051  genpdm  10939  isofn  17659  brric  20179  mpocti  31635  f1od2  31641  cnre2csqima  32495  elrnmpoid  43456  smflimlem3  45021  smflimlem6  45024
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