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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):AontoC (F B):BontoD) → (F (A B)):(A B)–1-1-onto→(C D))

Proof of Theorem resdif
StepHypRef Expression
1 fofun 5270 . . . . . 6 ((F A):AontoC → Fun (F A))
2 difss 3393 . . . . . . 7 (A B) A
3 fof 5269 . . . . . . . 8 ((F A):AontoC → (F A):A–→C)
4 fdm 5226 . . . . . . . 8 ((F A):A–→C → dom (F A) = A)
53, 4syl 15 . . . . . . 7 ((F A):AontoC → dom (F A) = A)
62, 5syl5sseqr 3320 . . . . . 6 ((F A):AontoC → (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)))
81, 6, 7syl2anc 642 . . . . 5 ((F A):AontoC → ((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)))
102, 9ax-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))))
1210, 11ax-mp 5 . . . . . 6 (((F A) (A B)):(A B)–onto→((F A) “ (A B)) ↔ (F (A B)):(A B)–onto→((F A) “ (A B)))
1310rneqi 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))
1613, 14, 153eqtr4i 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))))
1816, 17ax-mp 5 . . . . . 6 ((F (A B)):(A B)–onto→((F A) “ (A B)) ↔ (F (A B)):(A B)–onto→(F “ (A B)))
1912, 18bitri 240 . . . . 5 (((F A) (A B)):(A B)–onto→((F A) “ (A B)) ↔ (F (A B)):(A B)–onto→(F “ (A B)))
208, 19sylib 188 . . . 4 ((F A):AontoC → (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))))
2322biimpri 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)))
2420, 21, 23syl2anr 464 . . 3 ((Fun F (F A):AontoC) → (F (A B)):(A B)–1-1-onto→(F “ (A B)))
25243adant3 975 . 2 ((Fun F (F A):AontoC (F B):BontoD) → (F (A B)):(A B)–1-1-onto→(F “ (A B)))
26 dfima3 4951 . . . . . . 7 (FA) = ran (F A)
27 forn 5272 . . . . . . 7 ((F A):AontoC → ran (F A) = C)
2826, 27syl5eq 2397 . . . . . 6 ((F A):AontoC → (FA) = C)
29 dfima3 4951 . . . . . . 7 (FB) = ran (F B)
30 forn 5272 . . . . . . 7 ((F B):BontoD → ran (F B) = D)
3129, 30syl5eq 2397 . . . . . 6 ((F B):BontoD → (FB) = D)
3228, 31anim12i 549 . . . . 5 (((F A):AontoC (F B):BontoD) → ((FA) = C (FB) = D))
33 imadif 5171 . . . . . 6 (Fun F → (F “ (A B)) = ((FA) (FB)))
34 difeq12 3380 . . . . . 6 (((FA) = C (FB) = D) → ((FA) (FB)) = (C D))
3533, 34sylan9eq 2405 . . . . 5 ((Fun F ((FA) = C (FB) = D)) → (F “ (A B)) = (C D))
3632, 35sylan2 460 . . . 4 ((Fun F ((F A):AontoC (F B):BontoD)) → (F “ (A B)) = (C D))
37363impb 1147 . . 3 ((Fun F (F A):AontoC (F B):BontoD) → (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)))
3937, 38syl 15 . 2 ((Fun F (F A):AontoC (F B):BontoD) → ((F (A B)):(A B)–1-1-onto→(F “ (A B)) ↔ (F (A B)):(A B)–1-1-onto→(C D)))
4025, 39mpbid 201 1 ((Fun F (F A):AontoC (F B):BontoD) → (F (A B)):(A B)–1-1-onto→(C D))
 Colors of variables: wff setvar class 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
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