Theorem foimacnv 5304

Description: A reverse version of f1imacnv 5303. (Contributed by Jeffrey Hankins, 16-Jul-2009.)

Assertion

Ref	Expression
foimacnv	⊢ ((F:A–onto→B ∧ C ⊆ B) → (F “ (◡F “ C)) = C)

Proof of Theorem foimacnv

Step	Hyp	Ref	Expression
1		resima 5007	. 2 ⊢ ((F ↾ (◡F “ C)) “ (◡F “ C)) = (F “ (◡F “ C))
2		fofun 5271	. . . . . 6 ⊢ (F:A–onto→B → Fun F)
3	2	adantr 451	. . . . 5 ⊢ ((F:A–onto→B ∧ C ⊆ B) → Fun F)
4		funcnvres2 5168	. . . . 5 ⊢ (Fun F → ◡(◡F ↾ C) = (F ↾ (◡F “ C)))
5	3, 4	syl 15	. . . 4 ⊢ ((F:A–onto→B ∧ C ⊆ B) → ◡(◡F ↾ C) = (F ↾ (◡F “ C)))
6	5	imaeq1d 4942	. . 3 ⊢ ((F:A–onto→B ∧ C ⊆ B) → (◡(◡F ↾ C) “ (◡F “ C)) = ((F ↾ (◡F “ C)) “ (◡F “ C)))
7		resss 4989	. . . . . . . . . . 11 ⊢ (◡F ↾ C) ⊆ ◡F
8		cnvss 4886	. . . . . . . . . . 11 ⊢ ((◡F ↾ C) ⊆ ◡F → ◡(◡F ↾ C) ⊆ ◡◡F)
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ ◡(◡F ↾ C) ⊆ ◡◡F
10		cnvcnv 5063	. . . . . . . . . 10 ⊢ ◡◡F = F
11	9, 10	sseqtri 3304	. . . . . . . . 9 ⊢ ◡(◡F ↾ C) ⊆ F
12		funss 5127	. . . . . . . . 9 ⊢ (◡(◡F ↾ C) ⊆ F → (Fun F → Fun ◡(◡F ↾ C)))
13	11, 2, 12	mpsyl 59	. . . . . . . 8 ⊢ (F:A–onto→B → Fun ◡(◡F ↾ C))
14	13	adantr 451	. . . . . . 7 ⊢ ((F:A–onto→B ∧ C ⊆ B) → Fun ◡(◡F ↾ C))
15		dfima3 4952	. . . . . . . 8 ⊢ (◡F “ C) = ran (◡F ↾ C)
16		dfrn4 4905	. . . . . . . 8 ⊢ ran (◡F ↾ C) = dom ◡(◡F ↾ C)
17	15, 16	eqtr2i 2374	. . . . . . 7 ⊢ dom ◡(◡F ↾ C) = (◡F “ C)
18	14, 17	jctir 524	. . . . . 6 ⊢ ((F:A–onto→B ∧ C ⊆ B) → (Fun ◡(◡F ↾ C) ∧ dom ◡(◡F ↾ C) = (◡F “ C)))
19		df-fn 4791	. . . . . 6 ⊢ (◡(◡F ↾ C) Fn (◡F “ C) ↔ (Fun ◡(◡F ↾ C) ∧ dom ◡(◡F ↾ C) = (◡F “ C)))
20	18, 19	sylibr 203	. . . . 5 ⊢ ((F:A–onto→B ∧ C ⊆ B) → ◡(◡F ↾ C) Fn (◡F “ C))
21		df-dm 4788	. . . . . 6 ⊢ dom (◡F ↾ C) = ran ◡(◡F ↾ C)
22		forn 5273	. . . . . . . . . 10 ⊢ (F:A–onto→B → ran F = B)
23	22	sseq2d 3300	. . . . . . . . 9 ⊢ (F:A–onto→B → (C ⊆ ran F ↔ C ⊆ B))
24	23	biimpar 471	. . . . . . . 8 ⊢ ((F:A–onto→B ∧ C ⊆ B) → C ⊆ ran F)
25		dfrn4 4905	. . . . . . . 8 ⊢ ran F = dom ◡F
26	24, 25	syl6sseq 3318	. . . . . . 7 ⊢ ((F:A–onto→B ∧ C ⊆ B) → C ⊆ dom ◡F)
27		ssdmres 4988	. . . . . . 7 ⊢ (C ⊆ dom ◡F ↔ dom (◡F ↾ C) = C)
28	26, 27	sylib 188	. . . . . 6 ⊢ ((F:A–onto→B ∧ C ⊆ B) → dom (◡F ↾ C) = C)
29	21, 28	syl5eqr 2399	. . . . 5 ⊢ ((F:A–onto→B ∧ C ⊆ B) → ran ◡(◡F ↾ C) = C)
30		df-fo 4794	. . . . 5 ⊢ (◡(◡F ↾ C):(◡F “ C)–onto→C ↔ (◡(◡F ↾ C) Fn (◡F “ C) ∧ ran ◡(◡F ↾ C) = C))
31	20, 29, 30	sylanbrc 645	. . . 4 ⊢ ((F:A–onto→B ∧ C ⊆ B) → ◡(◡F ↾ C):(◡F “ C)–onto→C)
32		foima 5275	. . . 4 ⊢ (◡(◡F ↾ C):(◡F “ C)–onto→C → (◡(◡F ↾ C) “ (◡F “ C)) = C)
33	31, 32	syl 15	. . 3 ⊢ ((F:A–onto→B ∧ C ⊆ B) → (◡(◡F ↾ C) “ (◡F “ C)) = C)
34	6, 33	eqtr3d 2387	. 2 ⊢ ((F:A–onto→B ∧ C ⊆ B) → ((F ↾ (◡F “ C)) “ (◡F “ C)) = C)
35	1, 34	syl5eqr 2399	1 ⊢ ((F:A–onto→B ∧ C ⊆ B) → (F “ (◡F “ C)) = C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776   Fn wfn 4777  –onto→wfo 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 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-fo 4794

This theorem is referenced by: (None)