Theorem f1imacnv 6153

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 5431	. 2 ⊢ ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = (◡𝐹 “ (𝐹 “ 𝐶))
2		df-f1 5893	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 480	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
4	3	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡𝐹)
5		funcnvres 5967	. . . . 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 5465	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)))
8		f1ores 6151	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9		f1ocnv 6149	. . . . 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 5476	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = ran ◡(𝐹 ↾ 𝐶)
12		f1odm 6141	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → dom ◡(𝐹 ↾ 𝐶) = (𝐹 “ 𝐶))
13	12	imaeq2d 5466	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)))
14		f1ofo 6144	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶)
15		forn 6118	. . . . . 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 2680	. . . 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 2658	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = 𝐶)
20	1, 19	syl5eqr 2670	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ⊆ wss 3574  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  f1opw2  6888  ssenen  8134  f1opwfi  8270  isf34lem3  9197  subggim  17708  gicsubgen  17721  cnt1  21154  basqtop  21514  tgqtop  21515  hmeoopn  21569  hmeocld  21570  hmeontr  21572  qtopf1  21619  f1otrg  25751  tpr2rico  29958  eulerpartlemmf  30437  ballotlemscr  30580  ballotlemrinv0  30594  cvmlift2lem9a  31285  grpokerinj  33692