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Theorem recexprlemelu 7585

Description: Membership in the upper 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) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

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

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

**Proof of Theorem recexprlemelu**
Step	Hyp	Ref	Expression
1		elex 2741	. 2 ⊢ (𝐶 ∈ (2^nd ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7327	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4663	. . . . . 6 ⊢ (𝑦 <_Q 𝐶 → (𝑦 ∈ Q ∧ 𝐶 ∈ Q))
4	3	simprd 113	. . . . 5 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ Q)
5		elex 2741	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ V)
7	6	adantr 274	. . 3 ⊢ ((𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1591	. 2 ⊢ (∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
9		breq2 3993	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑦 <_Q 𝑥 ↔ 𝑦 <_Q 𝐶))
10	9	anbi1d 462	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) ↔ (𝑦 <_Q 𝐶 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
11	10	exbidv 1818	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (_Q‘𝑦) ∈ (1^st ‘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 5499	. . . 4 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘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) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1591	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3222	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4127	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘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) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1591	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3222	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4127	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op2nd 6126	. . . 4 ⊢ (2^nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
28	13, 27	eqtri 2191	. . 3 ⊢ (2^nd ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
29	11, 28	elab2g 2877	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))))
30	1, 8, 29	pm5.21nii 699	1 ⊢ (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))

Colors of variables: wff set class

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-2nd 6120 df-qs 6519 df-ni 7266 df-nqqs 7310 df-ltnqqs 7315

This theorem is referenced by: recexprlemm 7586 recexprlemopu 7589 recexprlemupu 7590 recexprlemdisj 7592 recexprlemloc 7593 recexprlem1ssu 7596 recexprlemss1u 7598

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