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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 5831 | . . . 4 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | 1st2nd 8064 | . . . 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 2842 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 6 | fvex 6919 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
| 7 | fvex 6919 | . . 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 5547 | . 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 1540 ∈ wcel 2108 〈cop 4632 {copab 5205 Rel wrel 5690 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: vciOLD 30580 sat1el2xp 35384 drngoi 37958 dicelval1sta 41189 |
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