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Theorem respreima 5238
 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 4874 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3120 . . . . . . . . 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 431 . . . . . . 7 ((x (B ∩ dom 𝐹) ((𝐹B)‘x) A) ↔ ((x dom 𝐹 x B) ((𝐹B)‘x) A))
6 fvres 5141 . . . . . . . . . 10 (x B → ((𝐹B)‘x) = (𝐹x))
76eleq1d 2103 . . . . . . . . 9 (x B → (((𝐹B)‘x) A ↔ (𝐹x) A))
87adantl 262 . . . . . . . 8 ((x dom 𝐹 x B) → (((𝐹B)‘x) A ↔ (𝐹x) A))
98pm5.32i 427 . . . . . . 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 496 . . . . 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 4939 . . . . . . . 8 (𝐹 Fn dom 𝐹 → Fun 𝐹)
15 funres 4884 . . . . . . . 8 (Fun 𝐹 → Fun (𝐹B))
1614, 15syl 14 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun (𝐹B))
17 dmres 4575 . . . . . . 7 dom (𝐹B) = (B ∩ dom 𝐹)
1816, 17jctir 296 . . . . . 6 (𝐹 Fn dom 𝐹 → (Fun (𝐹B) dom (𝐹B) = (B ∩ dom 𝐹)))
19 df-fn 4848 . . . . . 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 5229 . . . . 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 3120 . . . . 5 (x ((𝐹A) ∩ B) ↔ (x (𝐹A) x B))
24 elpreima 5229 . . . . . 6 (𝐹 Fn dom 𝐹 → (x (𝐹A) ↔ (x dom 𝐹 (𝐹x) A)))
2524anbi1d 438 . . . . 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 2035 1 (Fun 𝐹 → ((𝐹B) “ A) = ((𝐹A) ∩ B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   ∩ cin 2910  ◡ccnv 4287  dom cdm 4288   ↾ cres 4290   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845 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 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-bndl 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-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-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  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-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853 This theorem is referenced by: (None)
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