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Theorem ltexprlemelu 7540

Description: Element in upper cut of the constructed difference. Lemma for ltexpri 7554. (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 5850	. . . . 5 ⊢ (𝑥 = 𝑟 → (𝑦 +_Q 𝑥) = (𝑦 +_Q 𝑟))
2	1	eleq1d 2235	. . . 4 ⊢ (𝑥 = 𝑟 → ((𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵) ↔ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵)))
3	2	anbi2d 460	. . 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 1813	. 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 5489	. . 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 7304	. . . . 5 ⊢ Q ∈ V
8	7	rabex 4126	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ V
9	7	rabex 4126	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ V
10	8, 9	op2nd 6115	. . 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 2186	. 2 ⊢ (2^nd ‘𝐶) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}
12	4, 11	elrab2 2885	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 103 ↔ wb 104 = wceq 1343 ∃wex 1480 ∈ wcel 2136 {crab 2448 ⟨cop 3579 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 +_Q cplq 7223

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-2nd 6109 df-qs 6507 df-ni 7245 df-nqqs 7289

This theorem is referenced by: ltexprlemm 7541 ltexprlemopu 7544 ltexprlemupu 7545 ltexprlemdisj 7547 ltexprlemloc 7548 ltexprlemfu 7552 ltexprlemru 7553

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