Theorem foov 6209

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 5829	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)))
2		fveq2 5675	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 6061	. . . . . . 7 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2285	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eqeq2d 2246	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹‘𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
6	5	rexxp 4904	. . . 4 ⊢ (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
7	6	ralbii 2550	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
8	7	anbi2i 457	. 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 184	1 ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∀wral 2522  ∃wrex 2523  ⟨cop 3697   × cxp 4752  ⟶wf 5353  –onto→wfo 5355  ‘cfv 5357  (class class class)co 6058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-ov 6061

This theorem is referenced by:  xpsff1o  13646  mndpfo  13735