Theorem f1ococnv2 5310

Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by set.mm contributors, 13-Dec-2003.)

Assertion

Ref	Expression
f1ococnv2	⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = ( I ↾ B))

Proof of Theorem f1ococnv2

Step	Hyp	Ref	Expression
1		f1ofun 5290	. . 3 ⊢ (F:A–1-1-onto→B → Fun F)
2		df-fun 4790	. . . 4 ⊢ (Fun F ↔ (F ∘ ◡F) ⊆ I )
3		iss 5001	. . . 4 ⊢ ((F ∘ ◡F) ⊆ I ↔ (F ∘ ◡F) = ( I ↾ dom (F ∘ ◡F)))
4	2, 3	bitri 240	. . 3 ⊢ (Fun F ↔ (F ∘ ◡F) = ( I ↾ dom (F ∘ ◡F)))
5	1, 4	sylib 188	. 2 ⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = ( I ↾ dom (F ∘ ◡F)))
6		df-dm 4788	. . . . . 6 ⊢ dom F = ran ◡F
7		dmcoeq 4975	. . . . . 6 ⊢ (dom F = ran ◡F → dom (F ∘ ◡F) = dom ◡F)
8	6, 7	ax-mp 5	. . . . 5 ⊢ dom (F ∘ ◡F) = dom ◡F
9		dfrn4 4905	. . . . 5 ⊢ ran F = dom ◡F
10	8, 9	eqtr4i 2376	. . . 4 ⊢ dom (F ∘ ◡F) = ran F
11		f1ofo 5294	. . . . 5 ⊢ (F:A–1-1-onto→B → F:A–onto→B)
12		forn 5273	. . . . 5 ⊢ (F:A–onto→B → ran F = B)
13	11, 12	syl 15	. . . 4 ⊢ (F:A–1-1-onto→B → ran F = B)
14	10, 13	syl5eq 2397	. . 3 ⊢ (F:A–1-1-onto→B → dom (F ∘ ◡F) = B)
15	14	reseq2d 4935	. 2 ⊢ (F:A–1-1-onto→B → ( I ↾ dom (F ∘ ◡F)) = ( I ↾ B))
16	5, 15	eqtrd 2385	1 ⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = ( I ↾ B))

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3258   ∘ ccom 4722   I cid 4764  ^◡ccnv 4772  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776  –onto→wfo 4780  –1-1-onto→wf1o 4781

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  f1ococnv1  5311  f1cocnv2  5314  enmap2lem5  6068  enmap1lem5  6074