Theorem recexprlemell 7584

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7600. (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 2741	. 2 ⊢ (𝐶 ∈ (1st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7327	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4663	. . . . . 6 ⊢ (𝐶 <Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 111	. . . . 5 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ Q)
5		elex 2741	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ V)
7	6	adantr 274	. . 3 ⊢ ((𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1591	. 2 ⊢ (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 3992	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 <Q 𝑦 ↔ 𝐶 <Q 𝑦))
10	9	anbi1d 462	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
11	10	exbidv 1818	. . 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 5499	. . . 4 ⊢ (1st ‘𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩)
14		nqex 7325	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4663	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16	15	simpld 111	. . . . . . . . 9 ⊢ (𝑥 <Q 𝑦 → 𝑥 ∈ Q)
17	16	adantr 274	. . . . . . . 8 ⊢ ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1591	. . . . . . 7 ⊢ (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3222	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4127	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ∈ V
21	2	brel 4663	. . . . . . . . . 10 ⊢ (𝑦 <Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
22	21	simprd 113	. . . . . . . . 9 ⊢ (𝑦 <Q 𝑥 → 𝑥 ∈ Q)
23	22	adantr 274	. . . . . . . 8 ⊢ ((𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1591	. . . . . . 7 ⊢ (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3222	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4127	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ∈ V
27	20, 26	op1st 6125	. . . 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 2191	. . 3 ⊢ (1st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
29	11, 28	elab2g 2877	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 699	1 ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  Vcvv 2730  ⟨cop 3586   class class class wbr 3989  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Qcnq 7242  *_Qcrq 7246   <_Q cltq 7247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-qs 6519  df-ni 7266  df-nqqs 7310  df-ltnqqs 7315

This theorem is referenced by:  recexprlemm  7586  recexprlemopl  7587  recexprlemlol  7588  recexprlemdisj  7592  recexprlemloc  7593  recexprlem1ssl  7595  recexprlemss1l  7597