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Mirrors > Home > ILE Home > Th. List > elopabi | GIF version |
Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
Ref | Expression |
---|---|
elopabi.1 | ⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) |
elopabi.2 | ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elopabi | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4666 | . . . 4 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1st2nd 6079 | . . . 4 ⊢ ((Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
3 | 1, 2 | mpan 420 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
4 | id 19 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | 3, 4 | eqeltrrd 2217 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
6 | 1stexg 6065 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (1st ‘𝐴) ∈ V) | |
7 | 2ndexg 6066 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (2nd ‘𝐴) ∈ V) | |
8 | elopabi.1 | . . . 4 ⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) | |
9 | elopabi.2 | . . . 4 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | opelopabg 4190 | . . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
11 | 6, 7, 10 | syl2anc 408 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
12 | 5, 11 | mpbid 146 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 〈cop 3530 {copab 3988 Rel wrel 4544 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: aprcl 8408 |
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