Theorem f1ocnvb 5518

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 5517	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1ocnv 5517	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡◡𝐹:𝐴–1-1-onto→𝐵)
3		dfrel2 5120	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
4		f1oeq1 5492	. . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
5	3, 4	sylbi 121	. . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
6	2, 5	imbitrid 154	. 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵–1-1-onto→𝐴 → 𝐹:𝐴–1-1-onto→𝐵))
7	1, 6	impbid2 143	1 ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  ^◡ccnv 4662  Rel wrel 4668  –1-1-onto→wf1o 5257

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265

This theorem is referenced by: (None)