Theorem foimacnv 6781

Description: A reverse version of f1imacnv 6780. (Contributed by Jeff Hankins, 16-Jul-2009.)

Assertion

Ref	Expression
foimacnv	⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

Proof of Theorem foimacnv

Step	Hyp	Ref	Expression
1		resima 5966	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 6737	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
3	2	adantr 480	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 6562	. . . . 5 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
5	3, 4	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
6	5	imaeq1d 6010	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 5952	. . . . . . . . . 10 ⊢ (◡𝐹 ↾ 𝐶) ⊆ ◡𝐹
8		cnvss 5815	. . . . . . . . . 10 ⊢ ((◡𝐹 ↾ 𝐶) ⊆ ◡𝐹 → ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹
10		cnvcnvss 6143	. . . . . . . . 9 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3945	. . . . . . . 8 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 6501	. . . . . . . 8 ⊢ (◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun ◡(◡𝐹 ↾ 𝐶)))
13	11, 2, 12	mpsyl 68	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → Fun ◡(◡𝐹 ↾ 𝐶))
14	13	adantr 480	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 5632	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 5630	. . . . . . 7 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2753	. . . . . 6 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18		df-fn 6485	. . . . . 6 ⊢ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ↔ (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
19	14, 17, 18	sylanblrc 590	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶))
20		dfdm4 5838	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
21		forn 6739	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
22	21	sseq2d 3968	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
23	22	biimpar 477	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
24		df-rn 5630	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
25	23, 24	sseqtrdi 3976	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
26		ssdmres 5964	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
27	25, 26	sylib 218	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	20, 27	eqtr3id 2778	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
29		df-fo 6488	. . . . 5 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 ↔ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ∧ ran ◡(◡𝐹 ↾ 𝐶) = 𝐶))
30	19, 28, 29	sylanbrc 583	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶)
31		foima 6741	. . . 4 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
32	30, 31	syl 17	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
33	6, 32	eqtr3d 2766	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	1, 33	eqtr3id 2778	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ⊆ wss 3903  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6476   Fn wfn 6477  –onto→wfo 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488

This theorem is referenced by:  f1opw2  7604  mptcnfimad  7921  imacosupp  8142  fopwdom  9002  f1opwfi  9246  enfin2i  10215  fin1a2lem7  10300  fsumss  15632  fprodss  15855  gicsubgen  19158  coe1mul2lem2  22152  cncmp  23277  cnconn  23307  qtoprest  23602  qtopomap  23603  qtopcmap  23604  hmeoimaf1o  23655  elfm3  23835  imasf1oxms  24375  mbfimaopnlem  25554  cvmsss2  35257  diaintclN  41047  dibintclN  41156  dihintcl  41333  lnmepi  43068  pwfi2f1o  43079  sge0f1o  46373  isubgr3stgrlem8  47967