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Theorem cnvresima 6062
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 1957 . . . 4 (∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
2 vex 3402 . . . . . . . 8 𝑠 ∈ V
32opelresi 5833 . . . . . . 7 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (𝑡𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
4 vex 3402 . . . . . . . 8 𝑡 ∈ V
52, 4opelcnv 5724 . . . . . . 7 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴))
62, 4opelcnv 5724 . . . . . . . 8 (⟨𝑠, 𝑡⟩ ∈ 𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
76anbi2ci 628 . . . . . . 7 ((⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴) ↔ (𝑡𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
83, 5, 73bitr4i 306 . . . . . 6 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
98bianass 642 . . . . 5 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
109exbii 1854 . . . 4 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
114elima3 5910 . . . . 5 (𝑡 ∈ (𝐹𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹))
1211anbi1i 627 . . . 4 ((𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
131, 10, 123bitr4i 306 . . 3 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
144elima3 5910 . . 3 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)))
15 elin 3859 . . 3 (𝑡 ∈ ((𝐹𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
1613, 14, 153bitr4i 306 . 2 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
1716eqriv 2735 1 ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wex 1786  wcel 2114  cin 3842  cop 4522  ccnv 5524  cres 5527  cima 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538
This theorem is referenced by:  fimacnvinrn  6849  ramub2  16450  ramub1lem2  16463  cnrest  22036  kgencn  22307  kgencn3  22309  xkoptsub  22405  qtopres  22449  qtoprest  22468  mbfid  24387  mbfres  24396  1stpreima  30614  2ndpreima  30615  gsumhashmul  30893  cvmsss2  32807  lmhmlnmsplit  40484
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