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Theorem respreima 5220
Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
respreima (Fun 𝐹 → ((𝐹B) “ A) = ((𝐹A) ∩ B))

Proof of Theorem respreima
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 funfn 4857 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3103 . . . . . . . . 9 (x (B ∩ dom 𝐹) ↔ (x B x dom 𝐹))
3 ancom 253 . . . . . . . . 9 ((x B x dom 𝐹) ↔ (x dom 𝐹 x B))
42, 3bitri 173 . . . . . . . 8 (x (B ∩ dom 𝐹) ↔ (x dom 𝐹 x B))
54anbi1i 434 . . . . . . 7 ((x (B ∩ dom 𝐹) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 x B) ((𝐹B)‘x) A))
6 fvres 5123 . . . . . . . . . 10 (x B → ((𝐹B)‘x) = (𝐹x))
76eleq1d 2088 . . . . . . . . 9 (x B → (((𝐹B)‘x) A ↔ (𝐹x) A))
87adantl 262 . . . . . . . 8 ((x dom 𝐹 x B) → (((𝐹B)‘x) A ↔ (𝐹x) A))
98pm5.32i 430 . . . . . . 7 (((x dom 𝐹 x B) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 x B) (𝐹x) A))
105, 9bitri 173 . . . . . 6 ((x (B ∩ dom 𝐹) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 x B) (𝐹x) A))
1110a1i 9 . . . . 5 (𝐹 Fn dom 𝐹 → ((x (B ∩ dom 𝐹) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 x B) (𝐹x) A)))
12 an32 484 . . . . 5 (((x dom 𝐹 x B) (𝐹x) A) ↔ ((x dom 𝐹 (𝐹x) A) x B))
1311, 12syl6bb 185 . . . 4 (𝐹 Fn dom 𝐹 → ((x (B ∩ dom 𝐹) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 (𝐹x) A) x B)))
14 fnfun 4922 . . . . . . . 8 (𝐹 Fn dom 𝐹 → Fun 𝐹)
15 funres 4867 . . . . . . . 8 (Fun 𝐹 → Fun (𝐹B))
1614, 15syl 14 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun (𝐹B))
17 dmres 4559 . . . . . . 7 dom (𝐹B) = (B ∩ dom 𝐹)
1816, 17jctir 296 . . . . . 6 (𝐹 Fn dom 𝐹 → (Fun (𝐹B) dom (𝐹B) = (B ∩ dom 𝐹)))
19 df-fn 4832 . . . . . 6 ((𝐹B) Fn (B ∩ dom 𝐹) ↔ (Fun (𝐹B) dom (𝐹B) = (B ∩ dom 𝐹)))
2018, 19sylibr 137 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹B) Fn (B ∩ dom 𝐹))
21 elpreima 5211 . . . . 5 ((𝐹B) Fn (B ∩ dom 𝐹) → (x ((𝐹B) “ A) ↔ (x (B ∩ dom 𝐹) ((𝐹B)‘x) A)))
2220, 21syl 14 . . . 4 (𝐹 Fn dom 𝐹 → (x ((𝐹B) “ A) ↔ (x (B ∩ dom 𝐹) ((𝐹B)‘x) A)))
23 elin 3103 . . . . 5 (x ((𝐹A) ∩ B) ↔ (x (𝐹A) x B))
24 elpreima 5211 . . . . . 6 (𝐹 Fn dom 𝐹 → (x (𝐹A) ↔ (x dom 𝐹 (𝐹x) A)))
2524anbi1d 441 . . . . 5 (𝐹 Fn dom 𝐹 → ((x (𝐹A) x B) ↔ ((x dom 𝐹 (𝐹x) A) x B)))
2623, 25syl5bb 181 . . . 4 (𝐹 Fn dom 𝐹 → (x ((𝐹A) ∩ B) ↔ ((x dom 𝐹 (𝐹x) A) x B)))
2713, 22, 263bitr4d 209 . . 3 (𝐹 Fn dom 𝐹 → (x ((𝐹B) “ A) ↔ x ((𝐹A) ∩ B)))
281, 27sylbi 114 . 2 (Fun 𝐹 → (x ((𝐹B) “ A) ↔ x ((𝐹A) ∩ B)))
2928eqrdv 2020 1 (Fun 𝐹 → ((𝐹B) “ A) = ((𝐹A) ∩ B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  cin 2893  ccnv 4271  dom cdm 4272  cres 4274  cima 4275  Fun wfun 4823   Fn wfn 4824  cfv 4829
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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