Theorem recexprlemell 7841

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7857. (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 2814	. 2 ⊢ (𝐶 ∈ (1st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7584	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4778	. . . . . 6 ⊢ (𝐶 <Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 112	. . . . 5 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ Q)
5		elex 2814	. . . . 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 1646	. 2 ⊢ (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 4091	. . . . 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 1873	. . 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 5642	. . . 4 ⊢ (1st ‘𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩)
14		nqex 7582	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4778	. . . . . . . . . 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 1646	. . . . . . 7 ⊢ (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3302	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4227	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ∈ V
21	2	brel 4778	. . . . . . . . . 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 1646	. . . . . . 7 ⊢ (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3302	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4227	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ∈ V
27	20, 26	op1st 6308	. . . 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 2252	. . 3 ⊢ (1st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
29	11, 28	elab2g 2953	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 711	1 ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  Vcvv 2802  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301  Qcnq 7499  *_Qcrq 7503   <_Q cltq 7504

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-qs 6707  df-ni 7523  df-nqqs 7567  df-ltnqqs 7572

This theorem is referenced by:  recexprlemm  7843  recexprlemopl  7844  recexprlemlol  7845  recexprlemdisj  7849  recexprlemloc  7850  recexprlem1ssl  7852  recexprlemss1l  7854