Theorem foimacnv 6839

Description: A reverse version of f1imacnv 6838. (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 6012	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 6794	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
3	2	adantr 486	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 6617	. . . . 5 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
5	3, 4	syl 18	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
6	5	imaeq1d 6059	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 5998	. . . . . . . . . 10 ⊢ (◡𝐹 ↾ 𝐶) ⊆ ◡𝐹
8		cnvss 5856	. . . . . . . . . 10 ⊢ ((◡𝐹 ↾ 𝐶) ⊆ ◡𝐹 → ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹
10		cnvcnvss 6191	. . . . . . . . 9 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3943	. . . . . . . 8 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 6556	. . . . . . . 8 ⊢ (◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun ◡(◡𝐹 ↾ 𝐶)))
13	11, 2, 12	mpsyl 69	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → Fun ◡(◡𝐹 ↾ 𝐶))
14	13	adantr 486	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 5672	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 5670	. . . . . . 7 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2786	. . . . . 6 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18		df-fn 6540	. . . . . 6 ⊢ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ↔ (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
19	14, 17, 18	sylanblrc 602	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶))
20		dfdm4 5883	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
21		forn 6796	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
22	21	sseq2d 3966	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
23	22	biimpar 483	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
24		df-rn 5670	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
25	23, 24	sseqtrdi 3974	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
26		ssdmres 6010	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
27	25, 26	sylib 221	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	20, 27	eqtr3id 2811	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
29		df-fo 6543	. . . . 5 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 ↔ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ∧ ran ◡(◡𝐹 ↾ 𝐶) = 𝐶))
30	19, 28, 29	sylanbrc 595	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶)
31		foima 6798	. . . 4 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
32	30, 31	syl 18	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
33	6, 32	eqtr3d 2799	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	1, 33	eqtr3id 2811	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3902  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535

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 2147  ax-9 2155  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543

This theorem is used by:  f1opw2  7672  mptcnfimad  7986  imacosupp  8210  fopwdom  9086  f1opwfi  9326  enfin2i  10326  fin1a2lem7  10411  fsumss  15813  fprodss  16039  gicsubgen  19407  coe1mul2lem2  22495  cncmp  23618  cnconn  23648  qtoprest  23944  qtopomap  23945  qtopcmap  23946  hmeoimaf1o  23997  elfm3  24177  imasf1oxms  24716  mbfimaopnlem  25884  cvmsss2  35840  diaintclN  41918  dibintclN  42027  dihintcl  42204  lnmepi  43913  pwfi2f1o  43924  sge0f1o  47197  isubgr3stgrlem8  48876