Theorem resdif 5522

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 5477	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → Fun (𝐹 ↾ 𝐴))
2		difss 3285	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		fof 5476	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ 𝐴):𝐴⟶𝐶)
4		fdm 5409	. . . . . . . 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	sseqtrrid 3230	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐴 ∖ 𝐵) ⊆ dom (𝐹 ↾ 𝐴))
7		fores 5486	. . . . . 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 411	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)))
9		resres 4954	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵)))
10		indif 3402	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
11	10	reseq2i 4939	. . . . . . . 8 ⊢ (𝐹 ↾ (𝐴 ∩ (𝐴 ∖ 𝐵))) = (𝐹 ↾ (𝐴 ∖ 𝐵))
12	9, 11	eqtri 2214	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = (𝐹 ↾ (𝐴 ∖ 𝐵))
13		foeq1 5472	. . . . . . 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 5	. . . . . 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 4890	. . . . . . . 8 ⊢ ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
16		df-ima 4672	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = ran ((𝐹 ↾ 𝐴) ↾ (𝐴 ∖ 𝐵))
17		df-ima 4672	. . . . . . . 8 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = ran (𝐹 ↾ (𝐴 ∖ 𝐵))
18	15, 16, 17	3eqtr4i 2224	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) = (𝐹 “ (𝐴 ∖ 𝐵))
19		foeq3 5474	. . . . . . 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 5	. . . . . 6 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→((𝐹 ↾ 𝐴) “ (𝐴 ∖ 𝐵)) ↔ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
21	14, 20	bitri 184	. . . . 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 122	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐹 “ (𝐴 ∖ 𝐵)))
23		funres11 5326	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ (𝐴 ∖ 𝐵)))
24		dff1o3 5506	. . . . 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 133	. . . 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 290	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐹 “ (𝐴 ∖ 𝐵)))
27	26	3adant3 1019	. 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 4672	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29		forn 5479	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → ran (𝐹 ↾ 𝐴) = 𝐶)
30	28, 29	eqtrid 2238	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 → (𝐹 “ 𝐴) = 𝐶)
31		df-ima 4672	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
32		forn 5479	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → ran (𝐹 ↾ 𝐵) = 𝐷)
33	31, 32	eqtrid 2238	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→𝐷 → (𝐹 “ 𝐵) = 𝐷)
34	30, 33	anim12i 338	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷))
35		imadif 5334	. . . . . 6 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
36		difeq12 3272	. . . . . 6 ⊢ (((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷) → ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = (𝐶 ∖ 𝐷))
37	35, 36	sylan9eq 2246	. . . . 5 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 “ 𝐴) = 𝐶 ∧ (𝐹 “ 𝐵) = 𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
38	34, 37	sylan2 286	. . . 4 ⊢ ((Fun ◡𝐹 ∧ ((𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷)) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
39	38	3impb 1201	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 “ (𝐴 ∖ 𝐵)) = (𝐶 ∖ 𝐷))
40		f1oeq3 5490	. . 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 147	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 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∖ cdif 3150   ∩ cin 3152   ⊆ wss 3153  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   ↾ cres 4661   “ cima 4662  Fun wfun 5248  ⟶wf 5250  –onto→wfo 5252  –1-1-onto→wf1o 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261

This theorem is referenced by:  dif1en  6935