Theorem resdif 5168

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 5127	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → Fun (𝐹 ↾ 𝐴))
2		difss 3098	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		fof 5126	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ 𝐴):𝐴⟶𝐶)
4		fdm 5070	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐶 → dom (𝐹 ↾ 𝐴) = 𝐴)
5	3, 4	syl 14	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → dom (𝐹 ↾ 𝐴) = 𝐴)
6	2, 5	syl5sseqr 3048	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴))
7		fores 5135	. . . . . 6 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
8	1, 6, 7	syl2anc 403	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
9		resres 4642	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵)))
10		indif 3207	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
11	10	reseq2i 4627	. . . . . . . 8 ⊢ (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵))) = (𝐹 ↾ (𝐴 ∖ 𝐵))
12	9, 11	eqtri 2101	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∖ 𝐵))
13		foeq1 5122	. . . . . . 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))))
14	12, 13	ax-mp 7	. . . . . 6 ⊢ (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
15	12	rneqi 4580	. . . . . . . 8 ⊢ ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
16		df-ima 4376	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵))
17		df-ima 4376	. . . . . . . 8 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
18	15, 16, 17	3eqtr4i 2111	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = (𝐹 “ (𝐴 ∖ 𝐵))
19		foeq3 5124	. . . . . . 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))))
20	18, 19	ax-mp 7	. . . . . 6 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
21	14, 20	bitri 182	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
22	8, 21	sylib 120	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
23		funres11 4991	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵)))
24		dff1o3 5152	. . . . 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))))
25	24	biimpri 131	. . . 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)))
26	22, 23, 25	syl2anr 284	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
27	26	3adant3 958	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
28		df-ima 4376	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29		forn 5129	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ran (𝐹 ↾ 𝐴) = 𝐶)
30	28, 29	syl5eq 2125	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 “ 𝐴) = 𝐶)
31		df-ima 4376	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
32		forn 5129	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → ran (𝐹 ↾ 𝐵) = 𝐷)
33	31, 32	syl5eq 2125	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → (𝐹 “ 𝐵) = 𝐷)
34	30, 33	anim12i 331	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷))
35		imadif 4999	. . . . . 6 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
36		difeq12 3085	. . . . . 6 ⊢ (((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷) → ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = (𝐶 ∖ 𝐷))
37	35, 36	sylan9eq 2133	. . . . 5 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
38	34, 37	sylan2 280	. . . 4 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
39	38	3impb 1134	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
40		f1oeq3 5139	. . 3 ⊢ ((𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷) → ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷)))
41	39, 40	syl 14	. 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)))
42	27, 41	mpbid 145	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   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∖ cdif 2970   ∩ cin 2972   ⊆ wss 2973  ^◡ccnv 4362  dom cdm 4363  ran crn 4364   ↾ cres 4365   “ cima 4366  Fun wfun 4916  ⟶wf 4918  –onto→wfo 4920  –1-1-onto→wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  dif1en  6364