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Theorem fnmpt2 5855
Description: Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypothesis
Ref Expression
fmpt2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
fnmpt2 (∀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 fnmpt2
StepHypRef Expression
1 elex 2583 . . . 4 (𝐶𝑉𝐶 ∈ V)
21ralimi 2401 . . 3 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 𝐶 ∈ V)
32ralimi 2401 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶 ∈ V)
4 fmpt2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
54fmpt2 5854 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹:(𝐴 × 𝐵)⟶V)
6 dffn2 5074 . . 3 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
75, 6bitr4i 180 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 ∈ V ↔ 𝐹 Fn (𝐴 × 𝐵))
83, 7sylib 131 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574   × cxp 4370   Fn wfn 4924  wf 4925  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795
This theorem is referenced by:  fnmpt2i  5857  f1od2  5883  divfnzn  8652  cnref1o  8679
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