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Theorem recexprlemelu 7399
Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7414. (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 2671 . 2 (𝐶 ∈ (2nd𝐵) → 𝐶 ∈ V)
2 ltrelnq 7141 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4561 . . . . . 6 (𝑦 <Q 𝐶 → (𝑦Q𝐶Q))
43simprd 113 . . . . 5 (𝑦 <Q 𝐶𝐶Q)
5 elex 2671 . . . . 5 (𝐶Q𝐶 ∈ V)
64, 5syl 14 . . . 4 (𝑦 <Q 𝐶𝐶 ∈ V)
76adantr 274 . . 3 ((𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
87exlimiv 1562 . 2 (∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
9 breq2 3903 . . . . 5 (𝑥 = 𝐶 → (𝑦 <Q 𝑥𝑦 <Q 𝐶))
109anbi1d 460 . . . 4 (𝑥 = 𝐶 → ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
1110exbidv 1781 . . 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 5392 . . . 4 (2nd𝐵) = (2nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩)
14 nqex 7139 . . . . . 6 Q ∈ V
152brel 4561 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
1615simpld 111 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
1716adantr 274 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1817exlimiv 1562 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1918abssi 3142 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
2014, 19ssexi 4036 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ V
212brel 4561 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
2221simprd 113 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
2322adantr 274 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2423exlimiv 1562 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2524abssi 3142 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
2614, 25ssexi 4036 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ V
2720, 26op2nd 6013 . . . 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 2138 . . 3 (2nd𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}
2911, 28elab2g 2804 . 2 (𝐶 ∈ V → (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
301, 8, 29pm5.21nii 678 1 (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  {cab 2103  Vcvv 2660  cop 3500   class class class wbr 3899  cfv 5093  1st c1st 6004  2nd c2nd 6005  Qcnq 7056  *Qcrq 7060   <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-ltnqqs 7129
This theorem is referenced by:  recexprlemm  7400  recexprlemopu  7403  recexprlemupu  7404  recexprlemdisj  7406  recexprlemloc  7407  recexprlem1ssu  7410  recexprlemss1u  7412
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