Theorem foimacnv 6850

Description: A reverse version of f1imacnv 6849. (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 6015	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 6806	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
3	2	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 6628	. . . . 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 6058	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 6006	. . . . . . . . . 10 ⊢ (◡𝐹 ↾ 𝐶) ⊆ ◡𝐹
8		cnvss 5872	. . . . . . . . . 10 ⊢ ((◡𝐹 ↾ 𝐶) ⊆ ◡𝐹 → ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ ◡◡𝐹
10		cnvcnvss 6193	. . . . . . . . 9 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3991	. . . . . . . 8 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 6567	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 5689	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 5687	. . . . . . 7 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2761	. . . . . 6 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18		df-fn 6546	. . . . . 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 5895	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
21		forn 6808	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
22	21	sseq2d 4014	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
23	22	biimpar 478	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
24		df-rn 5687	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
25	23, 24	sseqtrdi 4032	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
26		ssdmres 6004	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
27	25, 26	sylib 217	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	20, 27	eqtr3id 2786	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
29		df-fo 6549	. . . . 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 6810	. . . 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 2774	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	1, 33	eqtr3id 2786	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (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   Fn wfn 6538  –onto→wfo 6541

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-fo 6549

This theorem is referenced by:  f1opw2  7663  imacosupp  8196  fopwdom  9082  f1opwfi  9358  enfin2i  10318  fin1a2lem7  10403  fsumss  15675  fprodss  15896  gicsubgen  19193  coe1mul2lem2  22010  cncmp  23116  cnconn  23146  qtoprest  23441  qtopomap  23442  qtopcmap  23443  hmeoimaf1o  23494  elfm3  23674  imasf1oxms  24218  mbfimaopnlem  25396  cvmsss2  34551  diaintclN  40232  dibintclN  40341  dihintcl  40518  lnmepi  42129  pwfi2f1o  42140  sge0f1o  45397