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Theorem foov 7073
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 6628 . 2 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)))
2 fveq2 6437 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 6913 . . . . . . 7 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2879 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eqeq2d 2835 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
65rexxp 5501 . . . 4 (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
76ralbii 3189 . . 3 (∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
87anbi2i 616 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
91, 8bitri 267 1 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1656  wral 3117  wrex 3118  cop 4405   × cxp 5344  wf 6123  ontowfo 6125  cfv 6127  (class class class)co 6910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-ov 6913
This theorem is referenced by:  iunfictbso  9257  xpsff1o  16588  mndpfo  17674  gafo  18086  isgrpo  27903  isgrpoi  27904  opidonOLD  34192  rngmgmbs4  34271  isgrpda  34295
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