MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  respreima Structured version   Visualization version   GIF version

Theorem respreima 7068
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 6579 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3965 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
32biancomi 464 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
43anbi1i 625 . . . . . . 7 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴))
5 fvres 6911 . . . . . . . . . 10 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
65eleq1d 2819 . . . . . . . . 9 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
76adantl 483 . . . . . . . 8 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
87pm5.32i 576 . . . . . . 7 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
94, 8bitri 275 . . . . . 6 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
109a1i 11 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴)))
11 an32 645 . . . . 5 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵))
1210, 11bitrdi 287 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
13 fnfun 6650 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun 𝐹)
1413funresd 6592 . . . . . 6 (𝐹 Fn dom 𝐹 → Fun (𝐹𝐵))
15 dmres 6004 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
16 df-fn 6547 . . . . . 6 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
1714, 15, 16sylanblrc 591 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹𝐵) Fn (𝐵 ∩ dom 𝐹))
18 elpreima 7060 . . . . 5 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
1917, 18syl 17 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
20 elin 3965 . . . . 5 (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵))
21 elpreima 7060 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
2221anbi1d 631 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2320, 22bitrid 283 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2412, 19, 233bitr4d 311 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
251, 24sylbi 216 . 2 (Fun 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
2625eqrdv 2731 1 (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3948  ccnv 5676  dom cdm 5677  cres 5679  cima 5680  Fun wfun 6538   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  paste  22798  restmetu  24079  eulerpartlemt  33370  smfres  45506
  Copyright terms: Public domain W3C validator