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)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		dffo3 7087	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)))
2		fveq2 6871	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7403	. . . . . . 7 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2818	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eqeq2d 2776	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹‘𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
6	5	rexxp 5818	. . . 4 ⊢ (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
7	6	ralbii 3111	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
8	7	anbi2i 634	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)))
9	1, 8	bitri 278	1 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563  ∀wral 3079  ∃wrex 3089  ⟨cop 4591   × cxp 5649  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-ov 7403

This theorem is referenced by:  iunfictbso  10086  xpsff1o  17609  mndpfo  18803  gafo  19354  isgrpo  30754  isgrpoi  30755  opidonOLD  38358  rngmgmbs4  38437  isgrpda  38461  ofoafo  43940  naddcnffo  43948