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Theorem resdif 5069
 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 5028 . . . . . 6 ((𝐹A):Aonto𝐶 → Fun (𝐹A))
2 difss 3043 . . . . . . 7 (AB) ⊆ A
3 fof 5027 . . . . . . . 8 ((𝐹A):Aonto𝐶 → (𝐹A):A𝐶)
4 fdm 4972 . . . . . . . 8 ((𝐹A):A𝐶 → dom (𝐹A) = A)
53, 4syl 14 . . . . . . 7 ((𝐹A):Aonto𝐶 → dom (𝐹A) = A)
62, 5syl5sseqr 2967 . . . . . 6 ((𝐹A):Aonto𝐶 → (AB) ⊆ dom (𝐹A))
7 fores 5036 . . . . . 6 ((Fun (𝐹A) (AB) ⊆ dom (𝐹A)) → ((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)))
81, 6, 7syl2anc 393 . . . . 5 ((𝐹A):Aonto𝐶 → ((𝐹A) ↾ (AB)):(AB)–onto→((𝐹A) “ (AB)))
9 resres 4547 . . . . . . . 8 ((𝐹A) ↾ (AB)) = (𝐹 ↾ (A ∩ (AB)))
10 indif 3153 . . . . . . . . 9 (A ∩ (AB)) = (AB)
1110reseq2i 4532 . . . . . . . 8 (𝐹 ↾ (A ∩ (AB))) = (𝐹 ↾ (AB))
129, 11eqtri 2038 . . . . . . 7 ((𝐹A) ↾ (AB)) = (𝐹 ↾ (AB))
13 foeq1 5023 . . . . . . 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 4485 . . . . . . . 8 ran ((𝐹A) ↾ (AB)) = ran (𝐹 ↾ (AB))
16 df-ima 4281 . . . . . . . 8 ((𝐹A) “ (AB)) = ran ((𝐹A) ↾ (AB))
17 df-ima 4281 . . . . . . . 8 (𝐹 “ (AB)) = ran (𝐹 ↾ (AB))
1815, 16, 173eqtr4i 2048 . . . . . . 7 ((𝐹A) “ (AB)) = (𝐹 “ (AB))
19 foeq3 5025 . . . . . . 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 4893 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (AB)))
24 dff1o3 5053 . . . . 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 910 . 2 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐹 “ (AB)))
28 df-ima 4281 . . . . . . 7 (𝐹A) = ran (𝐹A)
29 forn 5030 . . . . . . 7 ((𝐹A):Aonto𝐶 → ran (𝐹A) = 𝐶)
3028, 29syl5eq 2062 . . . . . 6 ((𝐹A):Aonto𝐶 → (𝐹A) = 𝐶)
31 df-ima 4281 . . . . . . 7 (𝐹B) = ran (𝐹B)
32 forn 5030 . . . . . . 7 ((𝐹B):Bonto𝐷 → ran (𝐹B) = 𝐷)
3331, 32syl5eq 2062 . . . . . 6 ((𝐹B):Bonto𝐷 → (𝐹B) = 𝐷)
3430, 33anim12i 321 . . . . 5 (((𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → ((𝐹A) = 𝐶 (𝐹B) = 𝐷))
35 imadif 4901 . . . . . 6 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∖ (𝐹B)))
36 difeq12 3030 . . . . . 6 (((𝐹A) = 𝐶 (𝐹B) = 𝐷) → ((𝐹A) ∖ (𝐹B)) = (𝐶𝐷))
3735, 36sylan9eq 2070 . . . . 5 ((Fun 𝐹 ((𝐹A) = 𝐶 (𝐹B) = 𝐷)) → (𝐹 “ (AB)) = (𝐶𝐷))
3834, 37sylan2 270 . . . 4 ((Fun 𝐹 ((𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷)) → (𝐹 “ (AB)) = (𝐶𝐷))
39383impb 1084 . . 3 ((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 “ (AB)) = (𝐶𝐷))
40 f1oeq3 5040 . . 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 871   = wceq 1226   ∖ cdif 2887   ∩ cin 2889   ⊆ wss 2890  ◡ccnv 4267  dom cdm 4268  ran crn 4269   ↾ cres 4270   “ cima 4271  Fun wfun 4819  ⟶wf 4821  –onto→wfo 4823  –1-1-onto→wf1o 4824 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832 This theorem is referenced by: (None)
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