Theorem f1ocnvb 6868

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

Assertion

Ref	Expression
f1ocnvb	⊢ (Rel 𝐹 → (𝐹:𝐴⟶1-1onto𝐵 ↔ ◡𝐹:𝐵⟶1-1onto𝐴))

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		f1ocnv 6867	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → ◡𝐹:𝐵⟶1-1onto𝐴)
2		f1ocnv 6867	. . 3 ⊢ (◡𝐹:𝐵⟶1-1onto𝐴 → ◡◡𝐹:𝐴⟶1-1onto𝐵)
3		dfrel2 6199	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
4		f1oeq1 6842	. . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴⟶1-1onto𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))
5	3, 4	sylbi 220	. . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴⟶1-1onto𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))
6	2, 5	imbitrid 247	. 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵⟶1-1onto𝐴 → 𝐹:𝐴⟶1-1onto𝐵))
7	1, 6	impbid2 229	1 ⊢ (Rel 𝐹 → (𝐹:𝐴⟶1-1onto𝐵 ↔ ◡𝐹:𝐵⟶1-1onto𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ^◡ccnv 5657  Rel wrel 5663  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  hasheqf1oi  14536