Theorem f1ocnvb 6846

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–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		f1ocnv 6845	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1ocnv 6845	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡◡𝐹:𝐴–1-1-onto→𝐵)
3		dfrel2 6188	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
4		f1oeq1 6821	. . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
5	3, 4	sylbi 216	. . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
6	2, 5	imbitrid 243	. 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵–1-1-onto→𝐴 → 𝐹:𝐴–1-1-onto→𝐵))
7	1, 6	impbid2 225	1 ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  ^◡ccnv 5675  Rel wrel 5681  –1-1-onto→wf1o 6542

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is referenced by:  hasheqf1oi  14310