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Theorem ltexprlemell 7078
 Description: Element in lower cut of the constructed difference. Lemma for ltexpri 7093. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlemell (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
Distinct variable groups:   𝑥,𝑦,𝑞,𝐴   𝑥,𝐵,𝑦,𝑞   𝑥,𝐶,𝑦,𝑞

Proof of Theorem ltexprlemell
StepHypRef Expression
1 oveq2 5602 . . . . 5 (𝑥 = 𝑞 → (𝑦 +Q 𝑥) = (𝑦 +Q 𝑞))
21eleq1d 2153 . . . 4 (𝑥 = 𝑞 → ((𝑦 +Q 𝑥) ∈ (1st𝐵) ↔ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
32anbi2d 452 . . 3 (𝑥 = 𝑞 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵)) ↔ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
43exbidv 1750 . 2 (𝑥 = 𝑞 → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5 ltexprlem.1 . . . 4 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
65fveq2i 5259 . . 3 (1st𝐶) = (1st ‘⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩)
7 nqex 6843 . . . . 5 Q ∈ V
87rabex 3951 . . . 4 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ∈ V
97rabex 3951 . . . 4 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ∈ V
108, 9op1st 5855 . . 3 (1st ‘⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩) = {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}
116, 10eqtri 2105 . 2 (1st𝐶) = {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}
124, 11elrab2 2764 1 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1287  ∃wex 1424   ∈ wcel 1436  {crab 2359  ⟨cop 3428  ‘cfv 4972  (class class class)co 5594  1st c1st 5847  2nd c2nd 5848  Qcnq 6760   +Q cplq 6762 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-iinf 4369 This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-1st 5849  df-qs 6231  df-ni 6784  df-nqqs 6828 This theorem is referenced by:  ltexprlemm  7080  ltexprlemopl  7081  ltexprlemlol  7082  ltexprlemdisj  7086  ltexprlemloc  7087  ltexprlemfl  7089  ltexprlemrl  7090
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