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Theorem recexprlemell 7755

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7771. (Contributed by Jim Kingdon, 27-Dec-2019.)

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

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

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

**Proof of Theorem recexprlemell**
Step	Hyp	Ref	Expression
1		elex 2785	. 2 ⊢ (𝐶 ∈ (1^st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7498	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4735	. . . . . 6 ⊢ (𝐶 <_Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 112	. . . . 5 ⊢ (𝐶 <_Q 𝑦 → 𝐶 ∈ Q)
5		elex 2785	. . . . 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) ∈ (2^nd ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1622	. 2 ⊢ (∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 4054	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 <_Q 𝑦 ↔ 𝐶 <_Q 𝑦))
10	9	anbi1d 465	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) ↔ (𝐶 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
11	10	exbidv 1849	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
12		recexpr.1	. . . . 5 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
13	12	fveq2i 5592	. . . 4 ⊢ (1^st ‘𝐵) = (1^st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩)
14		nqex 7496	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4735	. . . . . . . . . 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) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1622	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3272	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4190	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ V
21	2	brel 4735	. . . . . . . . . 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) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1622	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3272	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4190	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op1st 6245	. . . 4 ⊢ (1^st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))}
28	13, 27	eqtri 2227	. . 3 ⊢ (1^st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))}
29	11, 28	elab2g 2924	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 706	1 ⊢ (𝐶 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1373 ∃wex 1516 ∈ wcel 2177 {cab 2192 Vcvv 2773 ⟨cop 3641 class class class wbr 4051 ‘cfv 5280 1^st c1st 6237 2^nd c2nd 6238 Qcnq 7413 *_Qcrq 7417 <_Q cltq 7418

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-iinf 4644

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-id 4348 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-1st 6239 df-qs 6639 df-ni 7437 df-nqqs 7481 df-ltnqqs 7486

This theorem is referenced by: recexprlemm 7757 recexprlemopl 7758 recexprlemlol 7759 recexprlemdisj 7763 recexprlemloc 7764 recexprlem1ssl 7766 recexprlemss1l 7768

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