Theorem f1ocnvb 5298

Description: A class is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by set.mm contributors, 8-Dec-2003.) (Modified by Scott Fenton, 17-Apr-2021.)

Assertion

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

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		cnvcnv 5062	. . . . 5 ⊢ ◡◡F = F
2	1	fneq1i 5178	. . . 4 ⊢ (◡◡F Fn A ↔ F Fn A)
3	2	anbi2i 675	. . 3 ⊢ ((◡F Fn B ∧ ◡◡F Fn A) ↔ (◡F Fn B ∧ F Fn A))
4		ancom 437	. . 3 ⊢ ((◡F Fn B ∧ F Fn A) ↔ (F Fn A ∧ ◡F Fn B))
5	3, 4	bitri 240	. 2 ⊢ ((◡F Fn B ∧ ◡◡F Fn A) ↔ (F Fn A ∧ ◡F Fn B))
6		dff1o4 5294	. 2 ⊢ (◡F:B–1-1-onto→A ↔ (◡F Fn B ∧ ◡◡F Fn A))
7		dff1o4 5294	. 2 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))
8	5, 6, 7	3bitr4ri 269	1 ⊢ (F:A–1-1-onto→B ↔ ◡F:B–1-1-onto→A)

Syntax hints:   ↔ wb 176   ∧ wa 358  ^◡ccnv 4771   Fn wfn 4776  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  f1ocnv  5299