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Theorem ffnov 7552
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 7133 . 2 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶))
2 fveq2 6901 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7427 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2784 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eleq1d 2811 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
65ralxp 5848 . . 3 (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
76anbi2i 621 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
81, 7bitri 274 1 (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  cop 4639   × cxp 5680   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427
This theorem is referenced by:  fovcld  7553  cantnfvalf  9708  axaddf  11188  axmulf  11189  mulnzcnf  11910  frmdplusg  18844  gass  19295  sylow2blem2  19619  matecl  22418  txdis1cn  23630  isxmet2d  24324  prdsmet  24367  imasdsf1olem  24370  imasf1oxmet  24372  imasf1omet  24373  xmetresbl  24434  comet  24513  tgqioo  24807  xrtgioo  24813  opnmblALT  25623  mpodvdsmulf1o  27222  dvdsmulf1o  27224  hhssabloilem  31194  pstmxmet  33712  xrge0pluscn  33755  mpomulnzcnf  36011  isbndx  37483  isbnd3  37485  isbnd3b  37486  prdsbnd  37494  isdrngo2  37659  aks6d1c6lem3  41870  clintopcllaw  47588
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