Theorem foimacnv 6861

Description: A reverse version of f1imacnv 6860. (Contributed by Jeff Hankins, 16-Jul-2009.)

Assertion

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

Proof of Theorem foimacnv

Step	Hyp	Ref	Expression
1		resima 6067	. 2 ⊢ ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = (𝐹 “ (◡𝐹 “ 𝐶))
2		fofun 6816	. . . . . 6 ⊢ (𝐹:𝐴⟶onto𝐵 → Fun 𝐹)
3	2	adantr 486	. . . . 5 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun 𝐹)
4		funcnvres2 6639	. . . . 5 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
5	3, 4	syl 18	. . . 4 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
6	5	imaeq1d 6062	. . 3 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)))
7		resss 6002	. . . . . . . . . 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 6197	. . . . . . . . 9 ⊢ ◡◡𝐹 ⊆ 𝐹
11	9, 10	sstri 3922	. . . . . . . 8 ⊢ ◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹
12		funss 6575	. . . . . . . 8 ⊢ (◡(◡𝐹 ↾ 𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun ◡(◡𝐹 ↾ 𝐶)))
13	11, 2, 12	mpsyl 69	. . . . . . 7 ⊢ (𝐹:𝐴⟶onto𝐵 → Fun ◡(◡𝐹 ↾ 𝐶))
14	13	adantr 486	. . . . . 6 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → Fun ◡(◡𝐹 ↾ 𝐶))
15		df-ima 5668	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
16		df-rn 5666	. . . . . . 7 ⊢ ran (◡𝐹 ↾ 𝐶) = dom ◡(◡𝐹 ↾ 𝐶)
17	15, 16	eqtr2i 2787	. . . . . 6 ⊢ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
18		df-fn 6558	. . . . . 6 ⊢ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ↔ (Fun ◡(◡𝐹 ↾ 𝐶) ∧ dom ◡(◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
19	14, 17, 18	sylanblrc 602	. . . . 5 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶))
20		dfdm4 5883	. . . . . 6 ⊢ dom (◡𝐹 ↾ 𝐶) = ran ◡(◡𝐹 ↾ 𝐶)
21		forn 6818	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
22	21	sseq2d 3945	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐶 ⊆ ran 𝐹 ↔ 𝐶 ⊆ 𝐵))
23	22	biimpar 483	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ ran 𝐹)
24		df-rn 5666	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
25	23, 24	sseqtrdi 3953	. . . . . . 7 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ dom ◡𝐹)
26		ssdmres 6015	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
27	25, 26	sylib 221	. . . . . 6 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
28	20, 27	eqtr3id 2812	. . . . 5 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → ran ◡(◡𝐹 ↾ 𝐶) = 𝐶)
29		df-fo 6561	. . . . 5 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)⟶onto𝐶 ↔ (◡(◡𝐹 ↾ 𝐶) Fn (◡𝐹 “ 𝐶) ∧ ran ◡(◡𝐹 ↾ 𝐶) = 𝐶))
30	19, 28, 29	sylanbrc 595	. . . 4 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)⟶onto𝐶)
31		foima 6820	. . . 4 ⊢ (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)⟶onto𝐶 → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
32	30, 31	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → (◡(◡𝐹 ↾ 𝐶) “ (◡𝐹 “ 𝐶)) = 𝐶)
33	6, 32	eqtr3d 2800	. 2 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ↾ (◡𝐹 “ 𝐶)) “ (◡𝐹 “ 𝐶)) = 𝐶)
34	1, 33	eqtr3id 2812	1 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3900  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6549   Fn wfn 6550  ⟶ontowfo 6553

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561

This theorem is used by:  f1opw2  7697  mptcnfimad  8019  imacosupp  8251  fopwdom  9129  f1opwfi  9370  enfin2i  10423  fin1a2lem7  10508  fsumss  15917  fprodss  16141  gicsubgen  19520  coe1mul2lem2  22614  cncmp  23737  cnconn  23767  qtoprest  24063  qtopomap  24064  qtopcmap  24065  hmeoimaf1o  24116  elfm3  24296  imasf1oxms  24835  mbfimaopnlem  26003  cvmsss2  36205  diaintclN  42306  dibintclN  42415  dihintcl  42592  lnmepi  44278  pwfi2f1o  44289  sge0f1o  47595  isubgr3stgrlem8  49274