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Theorem cnvresima 5868
 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 2048 . . . 4 (∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
2 vex 3417 . . . . . . . 8 𝑠 ∈ V
32opelresi 5641 . . . . . . 7 (⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴) ↔ (𝑡𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
4 vex 3417 . . . . . . . 8 𝑡 ∈ V
52, 4opelcnv 5540 . . . . . . 7 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹𝐴))
62, 4opelcnv 5540 . . . . . . . 8 (⟨𝑠, 𝑡⟩ ∈ 𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
76anbi2ci 618 . . . . . . 7 ((⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴) ↔ (𝑡𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
83, 5, 73bitr4i 295 . . . . . 6 (⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ 𝐹𝑡𝐴))
98bianass 632 . . . . 5 ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
109exbii 1947 . . . 4 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ ∃𝑠((𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
114elima3 5718 . . . . 5 (𝑡 ∈ (𝐹𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹))
1211anbi1i 617 . . . 4 ((𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴) ↔ (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ 𝐹) ∧ 𝑡𝐴))
131, 10, 123bitr4i 295 . . 3 (∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
144elima3 5718 . . 3 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ ∃𝑠(𝑠𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ (𝐹𝐴)))
15 elin 4025 . . 3 (𝑡 ∈ ((𝐹𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (𝐹𝐵) ∧ 𝑡𝐴))
1613, 14, 153bitr4i 295 . 2 (𝑡 ∈ ((𝐹𝐴) “ 𝐵) ↔ 𝑡 ∈ ((𝐹𝐵) ∩ 𝐴))
1716eqriv 2822 1 ((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164   ∩ cin 3797  ⟨cop 4405  ◡ccnv 5345   ↾ cres 5348   “ cima 5349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359 This theorem is referenced by:  fimacnvinrn  6602  ramub2  16096  ramub1lem2  16109  cnrest  21467  kgencn  21737  kgencn3  21739  xkoptsub  21835  qtopres  21879  qtoprest  21898  mbfid  23808  mbfres  23817  1stpreima  30028  2ndpreima  30029  cvmsss2  31798  lmhmlnmsplit  38499
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