Theorem f1ocnvb 2812

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.

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 2811	. . 3 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)
2	1	a1i 7	. 2 ⊢ (Rel F → (F:A–1-1-onto→B → ◡F:B–1-1-onto→A))
3		dfrel2 2660	. . . 4 ⊢ (Rel F ↔ ◡◡F = F)
4		f1oeq1 2795	. . . 4 ⊢ (◡◡F = F → (◡◡F:A–1-1-onto→B ↔ F:A–1-1-onto→B))
5	3, 4	sylbi 174	. . 3 ⊢ (Rel F → (◡◡F:A–1-1-onto→B ↔ F:A–1-1-onto→B))
6		f1ocnv 2811	. . 3 ⊢ (◡F:B–1-1-onto→A → ◡◡F:A–1-1-onto→B)
7	5, 6	syl5bi 183	. 2 ⊢ (Rel F → (◡F:B–1-1-onto→A → F:A–1-1-onto→B))
8	2, 7	impbid 397	1 ⊢ (Rel F → (F:A–1-1-onto→B ↔ ◡F:B–1-1-onto→A))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091  ^◡ccnv 2409  Rel wrel 2415  –1-1-onto→wf1o 2421

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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