Theorem resdif 5306

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 5270	. . . . . 6 ⊢ ((F ↾ A):A–onto→C → Fun (F ↾ A))
2		difss 3393	. . . . . . 7 ⊢ (A ∖ B) ⊆ A
3		fof 5269	. . . . . . . 8 ⊢ ((F ↾ A):A–onto→C → (F ↾ A):A–→C)
4		fdm 5226	. . . . . . . 8 ⊢ ((F ↾ A):A–→C → dom (F ↾ A) = A)
5	3, 4	syl 15	. . . . . . 7 ⊢ ((F ↾ A):A–onto→C → dom (F ↾ A) = A)
6	2, 5	syl5sseqr 3320	. . . . . 6 ⊢ ((F ↾ A):A–onto→C → (A ∖ B) ⊆ dom (F ↾ A))
7		fores 5278	. . . . . 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 642	. . . . 5 ⊢ ((F ↾ A):A–onto→C → ((F ↾ A) ↾ (A ∖ B)):(A ∖ B)–onto→((F ↾ A) “ (A ∖ B)))
9		resabs1 4992	. . . . . . . 8 ⊢ ((A ∖ B) ⊆ A → ((F ↾ A) ↾ (A ∖ B)) = (F ↾ (A ∖ B)))
10	2, 9	ax-mp 5	. . . . . . 7 ⊢ ((F ↾ A) ↾ (A ∖ B)) = (F ↾ (A ∖ B))
11		foeq1 5265	. . . . . . 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))))
12	10, 11	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)))
13	10	rneqi 4957	. . . . . . . 8 ⊢ ran ((F ↾ A) ↾ (A ∖ B)) = ran (F ↾ (A ∖ B))
14		dfima3 4951	. . . . . . . 8 ⊢ ((F ↾ A) “ (A ∖ B)) = ran ((F ↾ A) ↾ (A ∖ B))
15		dfima3 4951	. . . . . . . 8 ⊢ (F “ (A ∖ B)) = ran (F ↾ (A ∖ B))
16	13, 14, 15	3eqtr4i 2383	. . . . . . 7 ⊢ ((F ↾ A) “ (A ∖ B)) = (F “ (A ∖ B))
17		foeq3 5267	. . . . . . 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))))
18	16, 17	ax-mp 5	. . . . . 6 ⊢ ((F ↾ (A ∖ B)):(A ∖ B)–onto→((F ↾ A) “ (A ∖ B)) ↔ (F ↾ (A ∖ B)):(A ∖ B)–onto→(F “ (A ∖ B)))
19	12, 18	bitri 240	. . . . 5 ⊢ (((F ↾ A) ↾ (A ∖ B)):(A ∖ B)–onto→((F ↾ A) “ (A ∖ B)) ↔ (F ↾ (A ∖ B)):(A ∖ B)–onto→(F “ (A ∖ B)))
20	8, 19	sylib 188	. . . 4 ⊢ ((F ↾ A):A–onto→C → (F ↾ (A ∖ B)):(A ∖ B)–onto→(F “ (A ∖ B)))
21		funres11 5164	. . . 4 ⊢ (Fun ◡F → Fun ◡(F ↾ (A ∖ B)))
22		dff1o3 5292	. . . . 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))))
23	22	biimpri 197	. . . 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)))
24	20, 21, 23	syl2anr 464	. . 3 ⊢ ((Fun ◡F ∧ (F ↾ A):A–onto→C) → (F ↾ (A ∖ B)):(A ∖ B)–1-1-onto→(F “ (A ∖ B)))
25	24	3adant3 975	. 2 ⊢ ((Fun ◡F ∧ (F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D) → (F ↾ (A ∖ B)):(A ∖ B)–1-1-onto→(F “ (A ∖ B)))
26		dfima3 4951	. . . . . . 7 ⊢ (F “ A) = ran (F ↾ A)
27		forn 5272	. . . . . . 7 ⊢ ((F ↾ A):A–onto→C → ran (F ↾ A) = C)
28	26, 27	syl5eq 2397	. . . . . 6 ⊢ ((F ↾ A):A–onto→C → (F “ A) = C)
29		dfima3 4951	. . . . . . 7 ⊢ (F “ B) = ran (F ↾ B)
30		forn 5272	. . . . . . 7 ⊢ ((F ↾ B):B–onto→D → ran (F ↾ B) = D)
31	29, 30	syl5eq 2397	. . . . . 6 ⊢ ((F ↾ B):B–onto→D → (F “ B) = D)
32	28, 31	anim12i 549	. . . . 5 ⊢ (((F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D) → ((F “ A) = C ∧ (F “ B) = D))
33		imadif 5171	. . . . . 6 ⊢ (Fun ◡F → (F “ (A ∖ B)) = ((F “ A) ∖ (F “ B)))
34		difeq12 3380	. . . . . 6 ⊢ (((F “ A) = C ∧ (F “ B) = D) → ((F “ A) ∖ (F “ B)) = (C ∖ D))
35	33, 34	sylan9eq 2405	. . . . 5 ⊢ ((Fun ◡F ∧ ((F “ A) = C ∧ (F “ B) = D)) → (F “ (A ∖ B)) = (C ∖ D))
36	32, 35	sylan2 460	. . . 4 ⊢ ((Fun ◡F ∧ ((F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D)) → (F “ (A ∖ B)) = (C ∖ D))
37	36	3impb 1147	. . 3 ⊢ ((Fun ◡F ∧ (F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D) → (F “ (A ∖ B)) = (C ∖ D))
38		f1oeq3 5283	. . 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)))
39	37, 38	syl 15	. 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	25, 39	mpbid 201	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 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∖ cdif 3206   ⊆ wss 3257   “ cima 4722  ^◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775  –→wf 4777  –onto→wfo 4779  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  resin  5307