Theorem dff1o6 7010

Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Assertion

Ref	Expression
dff1o6	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff1o6

Step	Hyp	Ref	Expression
1		df-f1o 6331	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		dff13 6991	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
3		df-fo 6330	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	2, 3	anbi12i 629	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5		df-3an 1086	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6		eqimss 3971	. . . . . . 7 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
7	6	anim2i 619	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6328	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
9	7, 8	sylibr 237	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
10	9	pm4.71ri 564	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
11	10	anbi1i 626	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
12		an32 645	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
13	5, 11, 12	3bitrri 301	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
14	1, 4, 13	3bitri 300	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∀wral 3106   ⊆ wss 3881  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by:  soisores  7059  f1otrg  26665  f1otrge  26666  grpoinvf  28315  bra11  29891  hgt750lemb  32037  diaf11N  38345  dibf11N  38457  lcfrlem9  38846  mapd1o  38944  hdmapf1oN  39161  hgmapf1oN  39199  rmxypairf1o  39852