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Theorem respreima 5624
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 5228 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3310 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
3 ancom 264 . . . . . . . . 9 ((𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
42, 3bitri 183 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
54anbi1i 455 . . . . . . 7 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴))
6 fvres 5520 . . . . . . . . . 10 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
76eleq1d 2239 . . . . . . . . 9 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
87adantl 275 . . . . . . . 8 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
98pm5.32i 451 . . . . . . 7 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
105, 9bitri 183 . . . . . 6 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
1110a1i 9 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴)))
12 an32 557 . . . . 5 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵))
1311, 12bitrdi 195 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
14 fnfun 5295 . . . . . . . 8 (𝐹 Fn dom 𝐹 → Fun 𝐹)
15 funres 5239 . . . . . . . 8 (Fun 𝐹 → Fun (𝐹𝐵))
1614, 15syl 14 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun (𝐹𝐵))
17 dmres 4912 . . . . . . 7 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1816, 17jctir 311 . . . . . 6 (𝐹 Fn dom 𝐹 → (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
19 df-fn 5201 . . . . . 6 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
2018, 19sylibr 133 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹𝐵) Fn (𝐵 ∩ dom 𝐹))
21 elpreima 5615 . . . . 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 3310 . . . . 5 (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵))
24 elpreima 5615 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
2524anbi1d 462 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2623, 25syl5bb 191 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2713, 22, 263bitr4d 219 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
281, 27sylbi 120 . 2 (Fun 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
2928eqrdv 2168 1 (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  cin 3120  ccnv 4610  dom cdm 4611  cres 4613  cima 4614  Fun wfun 5192   Fn wfn 5193  cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by: (None)
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