Theorem foov 7566

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 7077	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)))
2		fveq2 6861	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7393	. . . . . . 7 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2783	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eqeq2d 2741	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹‘𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
6	5	rexxp 5809	. . . 4 ⊢ (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
7	6	ralbii 3076	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
8	7	anbi2i 623	. 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 275	1 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∀wral 3045  ∃wrex 3054  ⟨cop 4598   × cxp 5639  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-ov 7393

This theorem is referenced by:  iunfictbso  10074  xpsff1o  17537  mndpfo  18691  gafo  19235  isgrpo  30433  isgrpoi  30434  opidonOLD  37853  rngmgmbs4  37932  isgrpda  37956  ofoafo  43352  naddcnffo  43360