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Theorem resdif 5091
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 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐶𝐷))

Proof of Theorem resdif
StepHypRef Expression
1 fofun 5050 . . . . . 6 ((𝐹A):Aonto𝐶 → Fun (𝐹A))
2 difss 3064 . . . . . . 7 (AB) ⊆ A
3 fof 5049 . . . . . . . 8 ((𝐹A):Aonto𝐶 → (𝐹A):A𝐶)
4 fdm 4993 . . . . . . . 8 ((𝐹A):A𝐶 → dom (𝐹A) = A)
53, 4syl 14 . . . . . . 7 ((𝐹A):Aonto𝐶 → dom (𝐹A) = A)
62, 5syl5sseqr 2988 . . . . . 6 ((𝐹A):Aonto𝐶 → (AB) ⊆ dom (𝐹A))
7 fores 5058 . . . . . 6 ((Fun (𝐹A) (AB) ⊆ dom (𝐹A)) → ((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)))
81, 6, 7syl2anc 391 . . . . 5 ((𝐹A):Aonto𝐶 → ((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)))
9 resres 4567 . . . . . . . 8 ((𝐹A) ↾ (AB)) = (𝐹 ↾ (A ∩ (AB)))
10 indif 3174 . . . . . . . . 9 (A ∩ (AB)) = (AB)
1110reseq2i 4552 . . . . . . . 8 (𝐹 ↾ (A ∩ (AB))) = (𝐹 ↾ (AB))
129, 11eqtri 2057 . . . . . . 7 ((𝐹A) ↾ (AB)) = (𝐹 ↾ (AB))
13 foeq1 5045 . . . . . . 7 (((𝐹A) ↾ (AB)) = (𝐹 ↾ (AB)) → (((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–onto→((𝐹A) “ (AB))))
1412, 13ax-mp 7 . . . . . 6 (((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)))
1512rneqi 4505 . . . . . . . 8 ran ((𝐹A) ↾ (AB)) = ran (𝐹 ↾ (AB))
16 df-ima 4301 . . . . . . . 8 ((𝐹A) “ (AB)) = ran ((𝐹A) ↾ (AB))
17 df-ima 4301 . . . . . . . 8 (𝐹 “ (AB)) = ran (𝐹 ↾ (AB))
1815, 16, 173eqtr4i 2067 . . . . . . 7 ((𝐹A) “ (AB)) = (𝐹 “ (AB))
19 foeq3 5047 . . . . . . 7 (((𝐹A) “ (AB)) = (𝐹 “ (AB)) → ((𝐹 ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB))))
2018, 19ax-mp 7 . . . . . 6 ((𝐹 ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB)))
2114, 20bitri 173 . . . . 5 (((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB)))
228, 21sylib 127 . . . 4 ((𝐹A):Aonto𝐶 → (𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB)))
23 funres11 4914 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (AB)))
24 dff1o3 5075 . . . . 5 ((𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)) ↔ ((𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB)) Fun (𝐹 ↾ (AB))))
2524biimpri 124 . . . 4 (((𝐹 ↾ (AB)):(AB)–onto→(𝐹 “ (AB)) Fun (𝐹 ↾ (AB))) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)))
2622, 23, 25syl2anr 274 . . 3 ((Fun 𝐹 (𝐹A):Aonto𝐶) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)))
27263adant3 923 . 2 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)))
28 df-ima 4301 . . . . . . 7 (𝐹A) = ran (𝐹A)
29 forn 5052 . . . . . . 7 ((𝐹A):Aonto𝐶 → ran (𝐹A) = 𝐶)
3028, 29syl5eq 2081 . . . . . 6 ((𝐹A):Aonto𝐶 → (𝐹A) = 𝐶)
31 df-ima 4301 . . . . . . 7 (𝐹B) = ran (𝐹B)
32 forn 5052 . . . . . . 7 ((𝐹B):Bonto𝐷 → ran (𝐹B) = 𝐷)
3331, 32syl5eq 2081 . . . . . 6 ((𝐹B):Bonto𝐷 → (𝐹B) = 𝐷)
3430, 33anim12i 321 . . . . 5 (((𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → ((𝐹A) = 𝐶 (𝐹B) = 𝐷))
35 imadif 4922 . . . . . 6 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∖ (𝐹B)))
36 difeq12 3051 . . . . . 6 (((𝐹A) = 𝐶 (𝐹B) = 𝐷) → ((𝐹A) ∖ (𝐹B)) = (𝐶𝐷))
3735, 36sylan9eq 2089 . . . . 5 ((Fun 𝐹 ((𝐹A) = 𝐶 (𝐹B) = 𝐷)) → (𝐹 “ (AB)) = (𝐶𝐷))
3834, 37sylan2 270 . . . 4 ((Fun 𝐹 ((𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷)) → (𝐹 “ (AB)) = (𝐶𝐷))
39383impb 1099 . . 3 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 “ (AB)) = (𝐶𝐷))
40 f1oeq3 5062 . . 3 ((𝐹 “ (AB)) = (𝐶𝐷) → ((𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐶𝐷)))
4139, 40syl 14 . 2 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → ((𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)) ↔ (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐶𝐷)))
4227, 41mpbid 135 1 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  cdif 2908  cin 2910  wss 2911  ccnv 4287  dom cdm 4288  ran crn 4289  cres 4290  cima 4291  Fun wfun 4839  wf 4841  ontowfo 4843  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by: (None)
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