Theorem resdif 6870

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 6822	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → Fun (𝐹 ↾ 𝐴))
2		difss 4146	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		fof 6821	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ 𝐴):𝐴⟶𝐶)
4	3	fdmd 6747	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → dom (𝐹 ↾ 𝐴) = 𝐴)
5	2, 4	sseqtrrid 4049	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴))
6		fores 6831	. . . . . 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 6013	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵)))
9		indif 4286	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
10	9	reseq2i 5997	. . . . . . . 8 ⊢ (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵))) = (𝐹 ↾ (𝐴 ∖ 𝐵))
11	8, 10	eqtri 2763	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∖ 𝐵))
12		foeq1 6817	. . . . . . 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 5951	. . . . . . . 8 ⊢ ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
15		df-ima 5702	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵))
16		df-ima 5702	. . . . . . . 8 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
17	14, 15, 16	3eqtr4i 2773	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = (𝐹 “ (𝐴 ∖ 𝐵))
18		foeq3 6819	. . . . . . 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 6645	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵)))
23		dff1o3 6855	. . . . 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 1131	. 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 5702	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
28		forn 6824	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ran (𝐹 ↾ 𝐴) = 𝐶)
29	27, 28	eqtrid 2787	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 “ 𝐴) = 𝐶)
30		df-ima 5702	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
31		forn 6824	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → ran (𝐹 ↾ 𝐵) = 𝐷)
32	30, 31	eqtrid 2787	. . . . . 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 6652	. . . . . 6 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
35		difeq12 4131	. . . . . 6 ⊢ (((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷) → ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = (𝐶 ∖ 𝐷))
36	34, 35	sylan9eq 2795	. . . . 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 6846	. 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 1537   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   ↾ cres 5691   “ cima 5692  Fun wfun 6557  –onto→wfo 6561  –1-1-onto→wf1o 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570

This theorem is referenced by:  resin  6871  dif1enlem  9195  dif1enlemOLD  9196  canthp1lem2  10691  symgcom  33086  cycpmconjvlem  33144  subfacp1lem3  35167  subfacp1lem5  35169