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Theorem recexprlemelu 7657
Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7672. (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 2763 . 2 (𝐶 ∈ (2nd𝐵) → 𝐶 ∈ V)
2 ltrelnq 7399 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4699 . . . . . 6 (𝑦 <Q 𝐶 → (𝑦Q𝐶Q))
43simprd 114 . . . . 5 (𝑦 <Q 𝐶𝐶Q)
5 elex 2763 . . . . 5 (𝐶Q𝐶 ∈ V)
64, 5syl 14 . . . 4 (𝑦 <Q 𝐶𝐶 ∈ V)
76adantr 276 . . 3 ((𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
87exlimiv 1609 . 2 (∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝐶 ∈ V)
9 breq2 4025 . . . . 5 (𝑥 = 𝐶 → (𝑦 <Q 𝑥𝑦 <Q 𝐶))
109anbi1d 465 . . . 4 (𝑥 = 𝐶 → ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
1110exbidv 1836 . . 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 5540 . . . 4 (2nd𝐵) = (2nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩)
14 nqex 7397 . . . . . 6 Q ∈ V
152brel 4699 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
1615simpld 112 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
1716adantr 276 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1817exlimiv 1609 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1918abssi 3245 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
2014, 19ssexi 4159 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ V
212brel 4699 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
2221simprd 114 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
2322adantr 276 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2423exlimiv 1609 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2524abssi 3245 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
2614, 25ssexi 4159 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ V
2720, 26op2nd 6176 . . . 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 2210 . . 3 (2nd𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}
2911, 28elab2g 2899 . 2 (𝐶 ∈ V → (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴))))
301, 8, 29pm5.21nii 705 1 (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  {cab 2175  Vcvv 2752  cop 3613   class class class wbr 4021  cfv 5238  1st c1st 6167  2nd c2nd 6168  Qcnq 7314  *Qcrq 7318   <Q cltq 7319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-iinf 4608
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-2nd 6170  df-qs 6569  df-ni 7338  df-nqqs 7382  df-ltnqqs 7387
This theorem is referenced by:  recexprlemm  7658  recexprlemopu  7661  recexprlemupu  7662  recexprlemdisj  7664  recexprlemloc  7665  recexprlem1ssu  7668  recexprlemss1u  7670
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