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Theorem ltexprlemelu 7666

Description: Element in upper cut of the constructed difference. Lemma for ltexpri 7680. (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 5930	. . . . 5 ⊢ (𝑥 = 𝑟 → (𝑦 +_Q 𝑥) = (𝑦 +_Q 𝑟))
2	1	eleq1d 2265	. . . 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 1839	. 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 5561	. . 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 7430	. . . . 5 ⊢ Q ∈ V
8	7	rabex 4177	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ V
9	7	rabex 4177	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ V
10	8, 9	op2nd 6205	. . 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 2217	. 2 ⊢ (2^nd ‘𝐶) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}
12	4, 11	elrab2 2923	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 1364 ∃wex 1506 ∈ wcel 2167 {crab 2479 ⟨cop 3625 ‘cfv 5258 (class class class)co 5922 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 +_Q cplq 7349

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-2nd 6199 df-qs 6598 df-ni 7371 df-nqqs 7415

This theorem is referenced by: ltexprlemm 7667 ltexprlemopu 7670 ltexprlemupu 7671 ltexprlemdisj 7673 ltexprlemloc 7674 ltexprlemfu 7678 ltexprlemru 7679

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