Theorem recexprlemell 7689

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7705. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemell	⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem recexprlemell

Step	Hyp	Ref	Expression
1		elex 2774	. 2 ⊢ (𝐶 ∈ (1st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7432	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4715	. . . . . 6 ⊢ (𝐶 <Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 112	. . . . 5 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ Q)
5		elex 2774	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ V)
7	6	adantr 276	. . 3 ⊢ ((𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1612	. 2 ⊢ (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 4036	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 <Q 𝑦 ↔ 𝐶 <Q 𝑦))
10	9	anbi1d 465	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
11	10	exbidv 1839	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
12		recexpr.1	. . . . 5 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
13	12	fveq2i 5561	. . . 4 ⊢ (1st ‘𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩)
14		nqex 7430	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4715	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16	15	simpld 112	. . . . . . . . 9 ⊢ (𝑥 <Q 𝑦 → 𝑥 ∈ Q)
17	16	adantr 276	. . . . . . . 8 ⊢ ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1612	. . . . . . 7 ⊢ (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3258	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4171	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ∈ V
21	2	brel 4715	. . . . . . . . . 10 ⊢ (𝑦 <Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
22	21	simprd 114	. . . . . . . . 9 ⊢ (𝑦 <Q 𝑥 → 𝑥 ∈ Q)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1612	. . . . . . 7 ⊢ (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3258	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4171	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ∈ V
27	20, 26	op1st 6204	. . . 4 ⊢ (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
28	13, 27	eqtri 2217	. . 3 ⊢ (1st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
29	11, 28	elab2g 2911	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 705	1 ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  Vcvv 2763  ⟨cop 3625   class class class wbr 4033  ‘cfv 5258  1^st c1st 6196  2^nd c2nd 6197  Qcnq 7347  *_Qcrq 7351   <_Q cltq 7352

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-qs 6598  df-ni 7371  df-nqqs 7415  df-ltnqqs 7420

This theorem is referenced by:  recexprlemm  7691  recexprlemopl  7692  recexprlemlol  7693  recexprlemdisj  7697  recexprlemloc  7698  recexprlem1ssl  7700  recexprlemss1l  7702