Theorem f1imacnv 6791

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 5975	. 2 ⊢ ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = (◡𝐹 “ (𝐹 “ 𝐶))
2		df-f1 6498	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 496	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
4	3	adantr 480	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡𝐹)
5		funcnvres 6571	. . . . 5 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶)))
6	4, 5	syl 17	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶)))
7	6	imaeq1d 6019	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)))
8		f1ores 6789	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9		f1ocnv 6787	. . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶)
11		imadmrn 6030	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = ran ◡(𝐹 ↾ 𝐶)
12		f1odm 6779	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → dom ◡(𝐹 ↾ 𝐶) = (𝐹 “ 𝐶))
13	12	imaeq2d 6020	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)))
14		f1ofo 6782	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶)
15		forn 6750	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶)
16	14, 15	syl 17	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶)
17	11, 13, 16	3eqtr3a 2796	. . . 4 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶)
18	10, 17	syl 17	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶)
19	7, 18	eqtr3d 2774	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = 𝐶)
20	1, 19	eqtr3id 2786	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ⊆ wss 3902  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Fun wfun 6487  ⟶wf 6489  –1-1→wf1 6490  –onto→wfo 6491  –1-1-onto→wf1o 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500

This theorem is referenced by:  f1opw2  7615  mptcnfimad  7932  ssenen  9083  f1opwfi  9260  isf34lem3  10289  subggim  19199  gicsubgen  19212  cnt1  23298  basqtop  23659  tgqtop  23660  hmeoopn  23714  hmeocld  23715  hmeontr  23717  qtopf1  23764  f1otrg  28926  tpr2rico  34050  eulerpartlemmf  34513  ballotlemscr  34657  ballotlemrinv0  34671  cvmlift2lem9a  35478  grpokerinj  38065  grimcnv  48170  uhgrimedg  48173  isubgr3stgrlem8  48255