Theorem f1imacnv 6841

Description: Preimage of an image. (Contributed by NM, 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 6016	. 2 ⊢ ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = (◡𝐹 “ (𝐹 “ 𝐶))
2		df-f1 6545	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 503	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
4	3	adantr 486	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡𝐹)
5		funcnvres 6618	. . . . 5 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶)))
6	4, 5	syl 18	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶)))
7	6	imaeq1d 6063	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)))
8		f1ores 6839	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9		f1ocnv 6837	. . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶)
10	8, 9	syl 18	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶)
11		imadmrn 6074	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = ran ◡(𝐹 ↾ 𝐶)
12		f1odm 6828	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → dom ◡(𝐹 ↾ 𝐶) = (𝐹 “ 𝐶))
13	12	imaeq2d 6064	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)))
14		f1ofo 6832	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶)
15		forn 6799	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶)
16	14, 15	syl 18	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶)
17	11, 13, 16	3eqtr3a 2824	. . . 4 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶)
18	10, 17	syl 18	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶)
19	7, 18	eqtr3d 2802	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = 𝐶)
20	1, 19	eqtr3id 2814	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Fun wfun 6534  ⟶wf 6536  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547

This theorem is used by:  f1opw2  7675  mptcnfimad  7989  ssenen  9146  f1opwfi  9320  isf34lem3  10374  subggim  19384  gicsubgen  19397  cnt1  23561  basqtop  23923  tgqtop  23924  hmeoopn  23978  hmeocld  23979  hmeontr  23981  qtopf1  24028  f1otrg  29279  tpr2rico  34370  eulerpartlemmf  34834  ballotlemscr  34978  ballotlemrinv0  34992  cvmlift2lem9a  35836  grpokerinj  38606  grimcnv  48730  uhgrimedg  48733  isubgr3stgrlem8  48815