Theorem foimacnv 5501

Description: A reverse version of f1imacnv 5500. (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 4961	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 5461	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
3	2	adantr 276	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 5313	. . . . 5 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
5	3, 4	syl 14	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
6	5	imaeq1d 4990	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 4952	. . . . . . . . . . 11 ⊢ (◡𝐹 ↾ 𝐶) ⊆ ◡𝐹
8		cnvss 4821	. . . . . . . . . . 11 ⊢ ((◡𝐹 ↾ 𝐶) ⊆ ◡𝐹 → ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹)
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹
10		cnvcnvss 5104	. . . . . . . . . 10 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3179	. . . . . . . . 9 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 5257	. . . . . . . . 9 ⊢ (◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun ◡(◡𝐹 ↾ 𝐶)))
13	11, 2, 12	mpsyl 65	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → Fun ◡(◡𝐹 ↾ 𝐶))
14	13	adantr 276	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 4660	. . . . . . . 8 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 4658	. . . . . . . 8 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2211	. . . . . . 7 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18	14, 17	jctir 313	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
19		df-fn 5241	. . . . . 6 ⊢ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ↔ (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
20	18, 19	sylibr 134	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶))
21		dfdm4 4840	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
22		forn 5463	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
23	22	sseq2d 3200	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
24	23	biimpar 297	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
25		df-rn 4658	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
26	24, 25	sseqtrdi 3218	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
27		ssdmres 4950	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	26, 27	sylib 122	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
29	21, 28	eqtr3id 2236	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
30		df-fo 5244	. . . . 5 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 ↔ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ∧ ran ◡(◡𝐹 ↾ 𝐶) = 𝐶))
31	20, 29, 30	sylanbrc 417	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶)
32		foima 5465	. . . 4 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
33	31, 32	syl 14	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	6, 33	eqtr3d 2224	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
35	1, 34	eqtr3id 2236	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ⊆ wss 3144  ^◡ccnv 4646  dom cdm 4647  ran crn 4648   ↾ cres 4649   “ cima 4650  Fun wfun 5232   Fn wfn 5233  –onto→wfo 5236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-fun 5240  df-fn 5241  df-f 5242  df-fo 5244

This theorem is referenced by:  f1opw2  6104  fopwdom  6868  fisumss  11441  fprodssdc  11639  hmeoimaf1o  14299