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Theorem ltexprlemelu 7821

Description: Element in upper cut of the constructed difference. Lemma for ltexpri 7835. (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 6028	. . . . 5 ⊢ (𝑥 = 𝑟 → (𝑦 +_Q 𝑥) = (𝑦 +_Q 𝑟))
2	1	eleq1d 2299	. . . 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 1872	. 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 5642	. . 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 7585	. . . . 5 ⊢ Q ∈ V
8	7	rabex 4233	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ V
9	7	rabex 4233	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ V
10	8, 9	op2nd 6312	. . 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 2251	. 2 ⊢ (2^nd ‘𝐶) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}
12	4, 11	elrab2 2964	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 1397 ∃wex 1540 ∈ wcel 2201 {crab 2513 ⟨cop 3671 ‘cfv 5325 (class class class)co 6020 1^st c1st 6303 2^nd c2nd 6304 Qcnq 7502 +_Q cplq 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 2203 ax-14 2204 ax-ext 2212 ax-coll 4203 ax-sep 4206 ax-pow 4263 ax-pr 4298 ax-un 4529 ax-iinf 4685

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ral 2514 df-rex 2515 df-reu 2516 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-pw 3653 df-sn 3674 df-pr 3675 df-op 3677 df-uni 3893 df-int 3928 df-iun 3971 df-br 4088 df-opab 4150 df-mpt 4151 df-id 4389 df-iom 4688 df-xp 4730 df-rel 4731 df-cnv 4732 df-co 4733 df-dm 4734 df-rn 4735 df-res 4736 df-ima 4737 df-iota 5285 df-fun 5327 df-fn 5328 df-f 5329 df-f1 5330 df-fo 5331 df-f1o 5332 df-fv 5333 df-ov 6023 df-2nd 6306 df-qs 6710 df-ni 7526 df-nqqs 7570

This theorem is referenced by: ltexprlemm 7822 ltexprlemopu 7825 ltexprlemupu 7826 ltexprlemdisj 7828 ltexprlemloc 7829 ltexprlemfu 7833 ltexprlemru 7834

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