Theorem foimacnv 6788

Description: A reverse version of f1imacnv 6787. (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 5974	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 6744	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
3	2	adantr 482	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 6569	. . . . 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 6018	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 5960	. . . . . . . . . 10 ⊢ (◡𝐹 ↾ 𝐶) ⊆ ◡𝐹
8		cnvss 5817	. . . . . . . . . 10 ⊢ ((◡𝐹 ↾ 𝐶) ⊆ ◡𝐹 → ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹
10		cnvcnvss 6149	. . . . . . . . 9 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3926	. . . . . . . 8 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 6508	. . . . . . . 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 482	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 5634	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 5632	. . . . . . 7 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2765	. . . . . 6 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18		df-fn 6492	. . . . . 6 ⊢ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ↔ (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
19	14, 17, 18	sylanblrc 597	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶))
20		dfdm4 5844	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
21		forn 6746	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
22	21	sseq2d 3949	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
23	22	biimpar 479	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
24		df-rn 5632	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
25	23, 24	sseqtrdi 3957	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
26		ssdmres 5972	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
27	25, 26	sylib 220	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	20, 27	eqtr3id 2790	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
29		df-fo 6495	. . . . 5 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶 ↔ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ∧ ran ◡(◡𝐹 ↾ 𝐶) = 𝐶))
30	19, 28, 29	sylanbrc 590	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–onto→𝐶)
31		foima 6748	. . . 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 2778	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	1, 33	eqtr3id 2790	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ⊆ wss 3885  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624  Fun wfun 6483   Fn wfn 6484  –onto→wfo 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495

This theorem is referenced by:  f1opw2  7615  mptcnfimad  7932  imacosupp  8153  fopwdom  9017  f1opwfi  9260  enfin2i  10238  fin1a2lem7  10323  fsumss  15682  fprodss  15908  gicsubgen  19249  coe1mul2lem2  22258  cncmp  23379  cnconn  23409  qtoprest  23704  qtopomap  23705  qtopcmap  23706  hmeoimaf1o  23757  elfm3  23937  imasf1oxms  24476  mbfimaopnlem  25644  cvmsss2  35517  diaintclN  41565  dibintclN  41674  dihintcl  41851  lnmepi  43545  pwfi2f1o  43556  sge0f1o  46839  isubgr3stgrlem8  48478