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Theorem recexprlemell 6593
 Description: Membership in the lower cut of B. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
Assertion
Ref Expression
recexprlemell (𝐶 (1stB) ↔ y(𝐶 <Q y (*Qy) (2ndA)))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem recexprlemell
StepHypRef Expression
1 elex 2560 . 2 (𝐶 (1stB) → 𝐶 V)
2 ltrelnq 6349 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4335 . . . . . 6 (𝐶 <Q y → (𝐶 Q y Q))
43simpld 105 . . . . 5 (𝐶 <Q y𝐶 Q)
5 elex 2560 . . . . 5 (𝐶 Q𝐶 V)
64, 5syl 14 . . . 4 (𝐶 <Q y𝐶 V)
76adantr 261 . . 3 ((𝐶 <Q y (*Qy) (2ndA)) → 𝐶 V)
87exlimiv 1486 . 2 (y(𝐶 <Q y (*Qy) (2ndA)) → 𝐶 V)
9 breq1 3758 . . . . 5 (x = 𝐶 → (x <Q y𝐶 <Q y))
109anbi1d 438 . . . 4 (x = 𝐶 → ((x <Q y (*Qy) (2ndA)) ↔ (𝐶 <Q y (*Qy) (2ndA))))
1110exbidv 1703 . . 3 (x = 𝐶 → (y(x <Q y (*Qy) (2ndA)) ↔ y(𝐶 <Q y (*Qy) (2ndA))))
12 recexpr.1 . . . . 5 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
1312fveq2i 5124 . . . 4 (1stB) = (1st ‘⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩)
14 nqex 6347 . . . . . 6 Q V
152brel 4335 . . . . . . . . . 10 (x <Q y → (x Q y Q))
1615simpld 105 . . . . . . . . 9 (x <Q yx Q)
1716adantr 261 . . . . . . . 8 ((x <Q y (*Qy) (2ndA)) → x Q)
1817exlimiv 1486 . . . . . . 7 (y(x <Q y (*Qy) (2ndA)) → x Q)
1918abssi 3009 . . . . . 6 {xy(x <Q y (*Qy) (2ndA))} ⊆ Q
2014, 19ssexi 3886 . . . . 5 {xy(x <Q y (*Qy) (2ndA))} V
212brel 4335 . . . . . . . . . 10 (y <Q x → (y Q x Q))
2221simprd 107 . . . . . . . . 9 (y <Q xx Q)
2322adantr 261 . . . . . . . 8 ((y <Q x (*Qy) (1stA)) → x Q)
2423exlimiv 1486 . . . . . . 7 (y(y <Q x (*Qy) (1stA)) → x Q)
2524abssi 3009 . . . . . 6 {xy(y <Q x (*Qy) (1stA))} ⊆ Q
2614, 25ssexi 3886 . . . . 5 {xy(y <Q x (*Qy) (1stA))} V
2720, 26op1st 5715 . . . 4 (1st ‘⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩) = {xy(x <Q y (*Qy) (2ndA))}
2813, 27eqtri 2057 . . 3 (1stB) = {xy(x <Q y (*Qy) (2ndA))}
2911, 28elab2g 2683 . 2 (𝐶 V → (𝐶 (1stB) ↔ y(𝐶 <Q y (*Qy) (2ndA))))
301, 8, 29pm5.21nii 619 1 (𝐶 (1stB) ↔ y(𝐶 <Q y (*Qy) (2ndA)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  *Qcrq 6268
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