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Theorem cnvresima 5090
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 vex 2727 . . . 4 𝑡 ∈ V
21elima3 4950 . . 3 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)))
31elima3 4950 . . . . 5 (𝑡 ∈ (𝐹𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹))
43anbi1i 454 . . . 4 ((𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
5 elin 3303 . . . 4 (𝑡 ∈ ((𝐹𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
6 vex 2727 . . . . . . . . . 10 𝑠 ∈ V
76, 1opelcnv 4783 . . . . . . . . 9 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴))
86opelres 4886 . . . . . . . . . 10 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
96, 1opelcnv 4783 . . . . . . . . . . 11 (⟨𝑠, 𝑡⟩ ∈ 𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
109anbi1i 454 . . . . . . . . . 10 ((⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
118, 10bitr4i 186 . . . . . . . . 9 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
127, 11bitri 183 . . . . . . . 8 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
1312anbi2i 453 . . . . . . 7 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (𝑠𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴)))
14 anass 399 . . . . . . 7 (((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (𝑠𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴)))
1513, 14bitr4i 186 . . . . . 6 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
1615exbii 1592 . . . . 5 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
17 19.41v 1889 . . . . 5 (∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
1816, 17bitri 183 . . . 4 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
194, 5, 183bitr4ri 212 . . 3 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
202, 19bitri 183 . 2 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
2120eqriv 2161 1 ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1342  wex 1479  wcel 2135  cin 3113  cop 3576  ccnv 4600  cres 4603  cima 4604
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-xp 4607  df-cnv 4609  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614
This theorem is referenced by:  cnrest  12833
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