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Theorem foov 7574
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 7093 . 2 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)))
2 fveq2 6881 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7404 . . . . . . 7 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2782 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eqeq2d 2735 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
65rexxp 5832 . . . 4 (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
76ralbii 3085 . . 3 (∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
87anbi2i 622 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
91, 8bitri 275 1 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wral 3053  wrex 3062  cop 4626   × cxp 5664  wf 6529  ontowfo 6531  cfv 6533  (class class class)co 7401
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-ov 7404
This theorem is referenced by:  iunfictbso  10105  xpsff1o  17512  mndpfo  18680  gafo  19202  isgrpo  30219  isgrpoi  30220  opidonOLD  37210  rngmgmbs4  37289  isgrpda  37313  ofoafo  42595  naddcnffo  42603
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