Theorem f1ocnvb 6369

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)

Assertion

Ref	Expression
f1ocnvb	⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		f1ocnv 6368	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1ocnv 6368	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡◡𝐹:𝐴–1-1-onto→𝐵)
3		dfrel2 5800	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
4		f1oeq1 6345	. . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
5	3, 4	sylbi 209	. . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
6	2, 5	syl5ib 236	. 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵–1-1-onto→𝐴 → 𝐹:𝐴–1-1-onto→𝐵))
7	1, 6	impbid2 218	1 ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653  ^◡ccnv 5311  Rel wrel 5317  –1-1-onto→wf1o 6100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108

This theorem is referenced by:  hasheqf1oi  13392