Theorem dff1o5 5509

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o5	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dff1o5

Step	Hyp	Ref	Expression
1		df-f1o 5261	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		dffo2 5480	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
3		f1f 5459	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
4	3	biantrurd 305	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)))
5	2, 4	bitr4id 199	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴–onto→𝐵 ↔ ran 𝐹 = 𝐵))
6	5	pm5.32i 454	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))
7	1, 6	bitri 184	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ran crn 4660  ⟶wf 5250  –1-1→wf1 5251  –onto→wfo 5252  –1-1-onto→wf1o 5253

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261

This theorem is referenced by:  f1orescnv  5516  f1finf1o  7006  djuinr  7122  eninl  7156  eninr  7157  frec2uzf1od  10477  ennnfonelemex  12571  ennnfonelemen  12578  ssnnctlemct  12603  pwf1oexmid  15490