Theorem foov 7595

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 7102	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)))
2		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7423	. . . . . . 7 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2813	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eqeq2d 2771	. . . . 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 3108	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
8	7	anbi2i 635	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∀wral 3076  ∃wrex 3086  ⟨cop 4834   × cxp 5648  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423

This theorem is used by:  iunfictbso  10196  xpsff1o  17744  mgmidpfod  18862  mndpfoOLD  18954  gafo  19515  isgrpo  31104  isgrpoi  31105  opidonOLD  38879  rngmgmbs4  38958  isgrpda  38982  ofoafo  44451  naddcnffo  44459