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Theorem ltexprlemelu 7914

Description: Element in upper cut of the constructed difference. Lemma for ltexpri 7928. (Contributed by Jim Kingdon, 21-Dec-2019.)

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

**Assertion**
Ref	Expression
ltexprlemelu	⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))

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

**Proof of Theorem ltexprlemelu**
Step	Hyp	Ref	Expression
1		oveq2 6058	. . . . 5 ⊢ (𝑥 = 𝑟 → (𝑦 +_Q 𝑥) = (𝑦 +_Q 𝑟))
2	1	eleq1d 2301	. . . 4 ⊢ (𝑥 = 𝑟 → ((𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵) ↔ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵)))
3	2	anbi2d 464	. . 3 ⊢ (𝑥 = 𝑟 → ((𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵)) ↔ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
4	3	exbidv 1874	. 2 ⊢ (𝑥 = 𝑟 → (∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵)) ↔ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
5		ltexprlem.1	. . . 4 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
6	5	fveq2i 5673	. . 3 ⊢ (2^nd ‘𝐶) = (2^nd ‘⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩)
7		nqex 7678	. . . . 5 ⊢ Q ∈ V
8	7	rabex 4256	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ V
9	7	rabex 4256	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ V
10	8, 9	op2nd 6341	. . 3 ⊢ (2^nd ‘⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}
11	6, 10	eqtri 2253	. 2 ⊢ (2^nd ‘𝐶) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}
12	4, 11	elrab2 2976	1 ⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1398 ∃wex 1541 ∈ wcel 2203 {crab 2524 ⟨cop 3692 ‘cfv 5352 (class class class)co 6050 1^st c1st 6332 2^nd c2nd 6333 Qcnq 7595 +_Q cplq 7597

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-iinf 4710

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-ov 6053 df-2nd 6335 df-qs 6773 df-ni 7619 df-nqqs 7663

This theorem is referenced by: ltexprlemm 7915 ltexprlemopu 7918 ltexprlemupu 7919 ltexprlemdisj 7921 ltexprlemloc 7922 ltexprlemfu 7926 ltexprlemru 7927

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