Theorem f1imacnv 5303

Description: Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.)

Assertion

Ref	Expression
f1imacnv	⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡F “ (F “ C)) = C)

Proof of Theorem f1imacnv

Step	Hyp	Ref	Expression
1		resima 5007	. 2 ⊢ ((◡F ↾ (F “ C)) “ (F “ C)) = (◡F “ (F “ C))
2		df-f1 4793	. . . . . 6 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
3	2	simprbi 450	. . . . 5 ⊢ (F:A–1-1→B → Fun ◡F)
4	3	adantr 451	. . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → Fun ◡F)
5		funcnvres 5166	. . . 4 ⊢ (Fun ◡F → ◡(F ↾ C) = (◡F ↾ (F “ C)))
6		imaeq1 4938	. . . 4 ⊢ (◡(F ↾ C) = (◡F ↾ (F “ C)) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C)))
7	4, 5, 6	3syl 18	. . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C)))
8		f1ores 5301	. . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C))
9		f1ocnv 5300	. . . 4 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → ◡(F ↾ C):(F “ C)–1-1-onto→C)
10		imadmrn 5009	. . . . . 6 ⊢ (◡(F ↾ C) “ dom ◡(F ↾ C)) = ran ◡(F ↾ C)
11		f1of 5288	. . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–→C)
12		fdm 5227	. . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–→C → dom ◡(F ↾ C) = (F “ C))
13		imaeq2 4939	. . . . . . 7 ⊢ (dom ◡(F ↾ C) = (F “ C) → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C)))
14	11, 12, 13	3syl 18	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C)))
15	10, 14	syl5reqr 2400	. . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = ran ◡(F ↾ C))
16		f1ofo 5294	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–onto→C)
17		forn 5273	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–onto→C → ran ◡(F ↾ C) = C)
18	16, 17	syl 15	. . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ran ◡(F ↾ C) = C)
19	15, 18	eqtrd 2385	. . . 4 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = C)
20	8, 9, 19	3syl 18	. . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = C)
21	7, 20	eqtr3d 2387	. 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → ((◡F ↾ (F “ C)) “ (F “ C)) = C)
22	1, 21	syl5eqr 2399	1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡F “ (F “ C)) = C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776  –→wf 4778  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781

This proof depends on 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 proof 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 used by: (None)