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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 4636 | . . . 4 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1st2nd 6047 | . . . 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 2195 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
6 | 1stexg 6033 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (1st ‘𝐴) ∈ V) | |
7 | 2ndexg 6034 | . . 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 4160 | . . 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 1316 ∈ wcel 1465 Vcvv 2660 〈cop 3500 {copab 3958 Rel wrel 4514 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: aprcl 8376 |
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