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Theorem cnvresima 4837
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 2577 . . . 4 𝑡 ∈ V
21elima3 4702 . . 3 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)))
31elima3 4702 . . . . 5 (𝑡 ∈ (𝐹𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹))
43anbi1i 439 . . . 4 ((𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
5 elin 3153 . . . 4 (𝑡 ∈ ((𝐹𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
6 vex 2577 . . . . . . . . . 10 𝑠 ∈ V
76, 1opelcnv 4544 . . . . . . . . 9 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴))
86opelres 4644 . . . . . . . . . 10 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
96, 1opelcnv 4544 . . . . . . . . . . 11 (⟨𝑠, 𝑡⟩ ∈ 𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
109anbi1i 439 . . . . . . . . . 10 ((⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹𝑡𝐴))
118, 10bitr4i 180 . . . . . . . . 9 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
127, 11bitri 177 . . . . . . . 8 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
1312anbi2i 438 . . . . . . 7 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (𝑠𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴)))
14 anass 387 . . . . . . 7 (((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (𝑠𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴)))
1513, 14bitr4i 180 . . . . . 6 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
1615exbii 1512 . . . . 5 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
17 19.41v 1798 . . . . 5 (∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
1816, 17bitri 177 . . . 4 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
194, 5, 183bitr4ri 206 . . 3 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
202, 19bitri 177 . 2 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
2120eqriv 2053 1 ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  cin 2943  cop 3405  ccnv 4371  cres 4374  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by: (None)
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