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Theorem recexprlemell 7278
Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7294. (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 2644 . 2 (𝐶 ∈ (1st𝐵) → 𝐶 ∈ V)
2 ltrelnq 7021 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4519 . . . . . 6 (𝐶 <Q 𝑦 → (𝐶Q𝑦Q))
43simpld 111 . . . . 5 (𝐶 <Q 𝑦𝐶Q)
5 elex 2644 . . . . 5 (𝐶Q𝐶 ∈ V)
64, 5syl 14 . . . 4 (𝐶 <Q 𝑦𝐶 ∈ V)
76adantr 271 . . 3 ((𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝐶 ∈ V)
87exlimiv 1541 . 2 (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝐶 ∈ V)
9 breq1 3870 . . . . 5 (𝑥 = 𝐶 → (𝑥 <Q 𝑦𝐶 <Q 𝑦))
109anbi1d 454 . . . 4 (𝑥 = 𝐶 → ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
1110exbidv 1760 . . 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 5343 . . . 4 (1st𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩)
14 nqex 7019 . . . . . 6 Q ∈ V
152brel 4519 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
1615simpld 111 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
1716adantr 271 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1817exlimiv 1541 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
1918abssi 3111 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
2014, 19ssexi 3998 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ V
212brel 4519 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
2221simprd 113 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
2322adantr 271 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2423exlimiv 1541 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
2524abssi 3111 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
2614, 25ssexi 3998 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ V
2720, 26op1st 5955 . . . 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 2115 . . 3 (1st𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}
2911, 28elab2g 2776 . 2 (𝐶 ∈ V → (𝐶 ∈ (1st𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
301, 8, 29pm5.21nii 658 1 (𝐶 ∈ (1st𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1296  wex 1433  wcel 1445  {cab 2081  Vcvv 2633  cop 3469   class class class wbr 3867  cfv 5049  1st c1st 5947  2nd c2nd 5948  Qcnq 6936  *Qcrq 6940   <Q cltq 6941
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-qs 6338  df-ni 6960  df-nqqs 7004  df-ltnqqs 7009
This theorem is referenced by:  recexprlemm  7280  recexprlemopl  7281  recexprlemlol  7282  recexprlemdisj  7286  recexprlemloc  7287  recexprlem1ssl  7289  recexprlemss1l  7291
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