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Theorem ffnov 7268
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 6878 . 2 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶))
2 fveq2 6667 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7151 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2879 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eleq1d 2902 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
65ralxp 5711 . . 3 (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
76anbi2i 622 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
81, 7bitri 276 1 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  cop 4570   × cxp 5552   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151
This theorem is referenced by:  fovcl  7269  cantnfvalf  9117  axaddf  10556  axmulf  10557  mulnzcnopr  11275  frmdplusg  18002  gass  18361  sylow2blem2  18666  matecl  20953  txdis1cn  22162  isxmet2d  22855  prdsmet  22898  imasdsf1olem  22901  imasf1oxmet  22903  imasf1omet  22904  xmetresbl  22965  comet  23041  tgqioo  23326  xrtgioo  23332  opnmblALT  24122  dvdsmulf1o  25688  hhssabloilem  28955  fovcld  30303  pstmxmet  31026  xrge0pluscn  31072  isbndx  34931  isbnd3  34933  isbnd3b  34934  prdsbnd  34942  isdrngo2  35107  clintopcllaw  43953
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