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Theorem ffnov 5976
Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
Assertion
Ref Expression
ffnov (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem ffnov
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ffnfv 5673 . 2 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶))
2 fveq2 5514 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 5875 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2228 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eleq1d 2246 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
65ralxp 4769 . . 3 (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
76anbi2i 457 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
81, 7bitri 184 1 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cop 3595   × cxp 4623   Fn wfn 5210  wf 5211  cfv 5215  (class class class)co 5872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875
This theorem is referenced by:  fovcl  5977  axaddf  7864  axmulf  7865  txdis1cn  13649  isxmet2d  13719  xmetresbl  13811  comet  13870  tgqioo  13918
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