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Theorem cnvresima 5541
Description: An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
Assertion
Ref Expression
cnvresima ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)

Proof of Theorem cnvresima
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.41v 1901 . . . 4 (∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
2 vex 3176 . . . . . . . 8 𝑠 ∈ V
32opelres 5322 . . . . . . 7 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
4 vex 3176 . . . . . . . 8 𝑡 ∈ V
52, 4opelcnv 5226 . . . . . . 7 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴))
62, 4opelcnv 5226 . . . . . . . 8 (⟨𝑠, 𝑡⟩ ∈ 𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
76anbi1i 727 . . . . . . 7 ((⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
83, 5, 73bitr4i 291 . . . . . 6 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
98bianass 838 . . . . 5 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
109exbii 1764 . . . 4 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
114elima3 5392 . . . . 5 (𝑡 ∈ (𝐹𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹))
1211anbi1i 727 . . . 4 ((𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
131, 10, 123bitr4i 291 . . 3 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
144elima3 5392 . . 3 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)))
15 elin 3758 . . 3 (𝑡 ∈ ((𝐹𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
1613, 14, 153bitr4i 291 . 2 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
1716eqriv 2607 1 ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wex 1695  wcel 1977  cin 3539  cop 4131  ccnv 5037  cres 5040  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by:  fimacnvinrn  6256  ramub2  15556  ramub1lem2  15569  cnrest  20899  kgencn  21169  kgencn3  21171  xkoptsub  21267  qtopres  21311  qtoprest  21330  mbfid  23209  mbfres  23217  1stpreima  28867  2ndpreima  28868  cvmsss2  30510  lmhmlnmsplit  36675
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