MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foov Structured version   Visualization version   GIF version

Theorem foov 7300
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 6844 . 2 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)))
2 fveq2 6646 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7136 . . . . . . 7 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2873 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eqeq2d 2831 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
65rexxp 5689 . . . 4 (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
76ralbii 3152 . . 3 (∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
87anbi2i 624 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
91, 8bitri 277 1 (𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wral 3125  wrex 3126  cop 4549   × cxp 5529  wf 6327  ontowfo 6329  cfv 6331  (class class class)co 7133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fo 6337  df-fv 6339  df-ov 7136
This theorem is referenced by:  iunfictbso  9518  xpsff1o  16819  mndpfo  17913  gafo  18405  isgrpo  28259  isgrpoi  28260  opidonOLD  35166  rngmgmbs4  35245  isgrpda  35269
  Copyright terms: Public domain W3C validator