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Theorem recexprlemelu 7438
Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7453. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemelu (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem recexprlemelu
StepHypRef Expression
1 elex 2697 . 2 (𝐶 ∈ (2nd𝐵) → 𝐶 ∈ V)
2 ltrelnq 7180 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4591 . . . . . 6 (𝑦 <Q 𝐶 → (𝑦Q𝐶Q))
43simprd 113 . . . . 5 (𝑦 <Q 𝐶𝐶Q)
5 elex 2697 . . . . 5 (𝐶Q𝐶 ∈ V)
64, 5syl 14 . . . 4 (𝑦 <Q 𝐶𝐶 ∈ V)
76adantr 274 . . 3 ((𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
87exlimiv 1577 . 2 (∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
9 breq2 3933 . . . . 5 (𝑥 = 𝐶 → (𝑦 <Q 𝑥𝑦 <Q 𝐶))
109anbi1d 460 . . . 4 (𝑥 = 𝐶 → ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
1110exbidv 1797 . . 3 (𝑥 = 𝐶 → (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
12 recexpr.1 . . . . 5 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
1312fveq2i 5424 . . . 4 (2nd𝐵) = (2nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩)
14 nqex 7178 . . . . . 6 Q ∈ V
152brel 4591 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
1615simpld 111 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
1716adantr 274 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1817exlimiv 1577 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1918abssi 3172 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
2014, 19ssexi 4066 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ V
212brel 4591 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
2221simprd 113 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
2322adantr 274 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2423exlimiv 1577 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2524abssi 3172 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
2614, 25ssexi 4066 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ V
2720, 26op2nd 6045 . . . 4 (2nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}
2813, 27eqtri 2160 . . 3 (2nd𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}
2911, 28elab2g 2831 . 2 (𝐶 ∈ V → (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
301, 8, 29pm5.21nii 693 1 (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  {cab 2125  Vcvv 2686  cop 3530   class class class wbr 3929  cfv 5123  1st c1st 6036  2nd c2nd 6037  Qcnq 7095  *Qcrq 7099   <Q cltq 7100
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-ltnqqs 7168
This theorem is referenced by:  recexprlemm  7439  recexprlemopu  7442  recexprlemupu  7443  recexprlemdisj  7445  recexprlemloc  7446  recexprlem1ssu  7449  recexprlemss1u  7451
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