Theorem resdif 6784

Description: The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
resdif	⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷))

Proof of Theorem resdif

Step	Hyp	Ref	Expression
1		fofun 6736	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → Fun (𝐹 ↾ 𝐴))
2		difss 4086	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		fof 6735	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ 𝐴):𝐴⟶𝐶)
4	3	fdmd 6661	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → dom (𝐹 ↾ 𝐴) = 𝐴)
5	2, 4	sseqtrrid 3978	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴))
6		fores 6745	. . . . . 6 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
7	1, 5, 6	syl2anc 584	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
8		resres 5941	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵)))
9		indif 4230	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
10	9	reseq2i 5925	. . . . . . . 8 ⊢ (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵))) = (𝐹 ↾ (𝐴 ∖ 𝐵))
11	8, 10	eqtri 2754	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∖ 𝐵))
12		foeq1 6731	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∖ 𝐵)) → (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵))))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
14	11	rneqi 5877	. . . . . . . 8 ⊢ ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
15		df-ima 5629	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵))
16		df-ima 5629	. . . . . . . 8 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
17	14, 15, 16	3eqtr4i 2764	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = (𝐹 “ (𝐴 ∖ 𝐵))
18		foeq3 6733	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = (𝐹 “ (𝐴 ∖ 𝐵)) → ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵))))
19	17, 18	ax-mp 5	. . . . . 6 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
20	13, 19	bitri 275	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
21	7, 20	sylib 218	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
22		funres11 6558	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵)))
23		dff1o3 6769	. . . . 5 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)) ↔ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)) ∧ Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵))))
24	23	biimpri 228	. . . 4 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)) ∧ Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵))) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
25	21, 22, 24	syl2anr 597	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
26	25	3adant3 1132	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
27		df-ima 5629	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
28		forn 6738	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ran (𝐹 ↾ 𝐴) = 𝐶)
29	27, 28	eqtrid 2778	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 “ 𝐴) = 𝐶)
30		df-ima 5629	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
31		forn 6738	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → ran (𝐹 ↾ 𝐵) = 𝐷)
32	30, 31	eqtrid 2778	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → (𝐹 “ 𝐵) = 𝐷)
33	29, 32	anim12i 613	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷))
34		imadif 6565	. . . . . 6 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
35		difeq12 4071	. . . . . 6 ⊢ (((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷) → ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = (𝐶 ∖ 𝐷))
36	34, 35	sylan9eq 2786	. . . . 5 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
37	33, 36	sylan2 593	. . . 4 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
38	37	3impb 1114	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
39	38	f1oeq3d 6760	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷)))
40	26, 39	mpbid 232	1 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ^◡ccnv 5615  dom cdm 5616  ran crn 5617   ↾ cres 5618   “ cima 5619  Fun wfun 6475  –onto→wfo 6479  –1-1-onto→wf1o 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488

This theorem is referenced by:  resin  6785  dif1enlem  9069  canthp1lem2  10541  symgcom  33047  cycpmconjvlem  33105  subfacp1lem3  35214  subfacp1lem5  35216