Theorem f1imacnv 2814

Description: Converse image of an image.

Assertion

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

Proof of Theorem f1imacnv

Step	Hyp	Ref	Expression
1		df-f1 2435	. . . . . 6 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
2	1	pm3.27bd 263	. . . . 5 ⊢ (F:A–1-1→B → Fun ◡F)
3	2	adantr 306	. . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → Fun ◡F)
4		funcnvres 2710	. . . 4 ⊢ (Fun ◡F → ◡(F ↾ C) = (◡F ↾ (F “ C)))
5		imaeq1 2602	. . . 4 ⊢ (◡(F ↾ C) = (◡F ↾ (F “ C)) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C)))
6	3, 4, 5	3syl 21	. . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C)))
7		f1ores 2813	. . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C))
8		f1ocnv 2811	. . . 4 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → ◡(F ↾ C):(F “ C)–1-1-onto→C)
9		f1of 2800	. . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–→C)
10		fdm 2756	. . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–→C → dom ◡(F ↾ C) = (F “ C))
11		imaeq2 2603	. . . . . . 7 ⊢ (dom ◡(F ↾ C) = (F “ C) → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C)))
12	9, 10, 11	3syl 21	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C)))
13		imadmrn 2610	. . . . . 6 ⊢ (◡(F ↾ C) “ dom ◡(F ↾ C)) = ran ◡(F ↾ C)
14	12, 13	syl5reqr 1139	. . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = ran ◡(F ↾ C))
15		f1ofo 2806	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–onto→C)
16		forn 2789	. . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–onto→C → ran ◡(F ↾ C) = C)
17	15, 16	syl 12	. . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ran ◡(F ↾ C) = C)
18	14, 17	eqtrd 1128	. . . 4 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = C)
19	7, 8, 18	3syl 21	. . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = C)
20	6, 19	eqtr3d 1130	. 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → ((◡F ↾ (F “ C)) “ (F “ C)) = C)
21		resima 2595	. 2 ⊢ ((◡F ↾ (F “ C)) “ (F “ C)) = (◡F “ (F “ C))
22	20, 21	syl5eqr 1138	1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡F “ (F “ C)) = C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  ssenen 3399

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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