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Mirrors > Home > MPE Home > Th. List > elopabi | Structured version Visualization version 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 | relopabv 5834 | . . . 4 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1st2nd 8063 | . . . 4 ⊢ ((Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
4 | id 22 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | 3, 4 | eqeltrrd 2840 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
6 | fvex 6920 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
7 | fvex 6920 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
8 | elopabi.1 | . . 3 ⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) | |
9 | elopabi.2 | . . 3 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) | |
10 | 6, 7, 8, 9 | opelopab 5552 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
11 | 5, 10 | sylib 218 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 〈cop 4637 {copab 5210 Rel wrel 5694 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: vciOLD 30590 sat1el2xp 35364 drngoi 37938 dicelval1sta 41170 |
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