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Mirrors > Home > ILE Home > Th. List > ltexprlemell | GIF version |
Description: Element in lower cut of the constructed difference. Lemma for ltexpri 7626. (Contributed by Jim Kingdon, 21-Dec-2019.) |
Ref | Expression |
---|---|
ltexprlem.1 | ⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 |
Ref | Expression |
---|---|
ltexprlemell | ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5896 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑦 +Q 𝑥) = (𝑦 +Q 𝑞)) | |
2 | 1 | eleq1d 2256 | . . . 4 ⊢ (𝑥 = 𝑞 → ((𝑦 +Q 𝑥) ∈ (1st ‘𝐵) ↔ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))) |
3 | 2 | anbi2d 464 | . . 3 ⊢ (𝑥 = 𝑞 → ((𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))) |
4 | 3 | exbidv 1835 | . 2 ⊢ (𝑥 = 𝑞 → (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵)) ↔ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))) |
5 | ltexprlem.1 | . . . 4 ⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 | |
6 | 5 | fveq2i 5530 | . . 3 ⊢ (1st ‘𝐶) = (1st ‘〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉) |
7 | nqex 7376 | . . . . 5 ⊢ Q ∈ V | |
8 | 7 | rabex 4159 | . . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))} ∈ V |
9 | 7 | rabex 4159 | . . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))} ∈ V |
10 | 8, 9 | op1st 6161 | . . 3 ⊢ (1st ‘〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))} |
11 | 6, 10 | eqtri 2208 | . 2 ⊢ (1st ‘𝐶) = {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))} |
12 | 4, 11 | elrab2 2908 | 1 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1363 ∃wex 1502 ∈ wcel 2158 {crab 2469 〈cop 3607 ‘cfv 5228 (class class class)co 5888 1st c1st 6153 2nd c2nd 6154 Qcnq 7293 +Q cplq 7295 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-1st 6155 df-qs 6555 df-ni 7317 df-nqqs 7361 |
This theorem is referenced by: ltexprlemm 7613 ltexprlemopl 7614 ltexprlemlol 7615 ltexprlemdisj 7619 ltexprlemloc 7620 ltexprlemfl 7622 ltexprlemrl 7623 |
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