Theorem f1imacnv 6849

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 6015	. 2 ⊢ ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = (◡𝐹 “ (𝐹 “ 𝐶))
2		df-f1 6548	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 497	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
4	3	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡𝐹)
5		funcnvres 6626	. . . . 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 6058	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)))
8		f1ores 6847	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9		f1ocnv 6845	. . . . 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 6069	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = ran ◡(𝐹 ↾ 𝐶)
12		f1odm 6837	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → dom ◡(𝐹 ↾ 𝐶) = (𝐹 “ 𝐶))
13	12	imaeq2d 6059	. . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)))
14		f1ofo 6840	. . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶)
15		forn 6808	. . . . . 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 396   = wceq 1541   ⊆ wss 3948  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ↾ cres 5678   “ cima 5679  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is referenced by:  f1opw2  7663  ssenen  9153  f1opwfi  9358  isf34lem3  10372  subggim  19142  gicsubgen  19154  cnt1  22861  basqtop  23222  tgqtop  23223  hmeoopn  23277  hmeocld  23278  hmeontr  23280  qtopf1  23327  f1otrg  28160  tpr2rico  32961  eulerpartlemmf  33443  ballotlemscr  33586  ballotlemrinv0  33600  cvmlift2lem9a  34363  grpokerinj  36847  isomgrsym  46583