Theorem dff1o4 5294

Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o4	⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))

Proof of Theorem dff1o4

Step	Hyp	Ref	Expression
1		dff1o2 5291	. 2 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))
2		3anass 938	. 2 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
3		dfrn4 4904	. . . . . 6 ⊢ ran F = dom ◡F
4	3	eqeq1i 2360	. . . . 5 ⊢ (ran F = B ↔ dom ◡F = B)
5	4	anbi2i 675	. . . 4 ⊢ ((Fun ◡F ∧ ran F = B) ↔ (Fun ◡F ∧ dom ◡F = B))
6		df-fn 4790	. . . 4 ⊢ (◡F Fn B ↔ (Fun ◡F ∧ dom ◡F = B))
7	5, 6	bitr4i 243	. . 3 ⊢ ((Fun ◡F ∧ ran F = B) ↔ ◡F Fn B)
8	7	anbi2i 675	. 2 ⊢ ((F Fn A ∧ (Fun ◡F ∧ ran F = B)) ↔ (F Fn A ∧ ◡F Fn B))
9	1, 2, 8	3bitri 262	1 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642  ^◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  f1ocnvb  5298  f1oun  5304  f1o00  5317  f1oi  5320  f1ovi  5321  f1osn  5322  swapf1o  5511  f1od  5726  f1opprod  5844  enex  6031  enpw1  6062  enmap2  6068  scancan  6331