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Theorem recexprlemell 7244
Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7260. (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
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 2633 . 2 (𝐶 ∈ (1st𝐵) → 𝐶 ∈ V)
2 ltrelnq 6987 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4505 . . . . . 6 (𝐶 <Q 𝑦 → (𝐶Q𝑦Q))
43simpld 111 . . . . 5 (𝐶 <Q 𝑦𝐶Q)
5 elex 2633 . . . . 5 (𝐶Q𝐶 ∈ V)
64, 5syl 14 . . . 4 (𝐶 <Q 𝑦𝐶 ∈ V)
76adantr 271 . . 3 ((𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝐶 ∈ V)
87exlimiv 1535 . 2 (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝐶 ∈ V)
9 breq1 3856 . . . . 5 (𝑥 = 𝐶 → (𝑥 <Q 𝑦𝐶 <Q 𝑦))
109anbi1d 454 . . . 4 (𝑥 = 𝐶 → ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
1110exbidv 1754 . . 3 (𝑥 = 𝐶 → (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
12 recexpr.1 . . . . 5 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
1312fveq2i 5323 . . . 4 (1st𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩)
14 nqex 6985 . . . . . 6 Q ∈ V
152brel 4505 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
1615simpld 111 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
1716adantr 271 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1817exlimiv 1535 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1918abssi 3099 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
2014, 19ssexi 3985 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ V
212brel 4505 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
2221simprd 113 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
2322adantr 271 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2423exlimiv 1535 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2524abssi 3099 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
2614, 25ssexi 3985 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ V
2720, 26op1st 5933 . . . 4 (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}
2813, 27eqtri 2109 . . 3 (1st𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}
2911, 28elab2g 2765 . 2 (𝐶 ∈ V → (𝐶 ∈ (1st𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
301, 8, 29pm5.21nii 656 1 (𝐶 ∈ (1st𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1290  wex 1427  wcel 1439  {cab 2075  Vcvv 2622  cop 3455   class class class wbr 3853  cfv 5030  1st c1st 5925  2nd c2nd 5926  Qcnq 6902  *Qcrq 6906   <Q cltq 6907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-iinf 4418
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-1st 5927  df-qs 6314  df-ni 6926  df-nqqs 6970  df-ltnqqs 6975
This theorem is referenced by:  recexprlemm  7246  recexprlemopl  7247  recexprlemlol  7248  recexprlemdisj  7252  recexprlemloc  7253  recexprlem1ssl  7255  recexprlemss1l  7257
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