Theorem dff1o5 5648

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 5384	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		dffo2 5619	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
3		f1f 5598	. . . . 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 458	. 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))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402  ran crn 4775  ⟶wf 5373  –1-1→wf1 5374  –onto→wfo 5375  –1-1-onto→wf1o 5376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384

This theorem is used by:  f1orescnv  5655  f1finf1o  7264  djuinr  7403  eninl  7437  eninr  7438  frec2uzf1od  10843  ennnfonelemex  13305  ennnfonelemen  13312  ssnnctlemct  13337  2lgslem1b  16208  ausgrusgrben  16409  pwf1oexmid  17029