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Theorem foov 7401
Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
foov (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶   𝑥,𝐹,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem foov
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dffo3 6940 . 2 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)))
2 fveq2 6736 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7235 . . . . . . 7 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2797 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eqeq2d 2749 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
65rexxp 5726 . . . 4 (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
76ralbii 3089 . . 3 (∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
87anbi2i 626 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
91, 8bitri 278 1 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wral 3062  wrex 3063  cop 4562   × cxp 5564  wf 6394  ontowfo 6396  cfv 6398  (class class class)co 7232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-fo 6404  df-fv 6406  df-ov 7235
This theorem is referenced by:  iunfictbso  9753  xpsff1o  17100  mndpfo  18224  gafo  18718  isgrpo  28605  isgrpoi  28606  opidonOLD  35777  rngmgmbs4  35856  isgrpda  35880
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