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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopab3 | Structured version Visualization version GIF version | ||
| Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.) | 
| Ref | Expression | 
|---|---|
| opelopab3.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| opelopab3.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | 
| opelopab3.3 | ⊢ (𝜒 → 𝐴 ∈ 𝐶) | 
| Ref | Expression | 
|---|---|
| opelopab3 | ⊢ (𝐵 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elopaelxp 5775 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
| 2 | opelxp1 5727 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) → 𝐴 ∈ V) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ V) | 
| 4 | 3 | anim1i 615 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) | 
| 5 | 4 | ancoms 458 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) | 
| 6 | opelopab3.3 | . . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶) | |
| 7 | 6 | elexd 3504 | . . . 4 ⊢ (𝜒 → 𝐴 ∈ V) | 
| 8 | 7 | anim1i 615 | . . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) | 
| 9 | 8 | ancoms 458 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) | 
| 10 | opelopab3.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 11 | opelopab3.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 12 | 10, 11 | opelopabg 5543 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | 
| 13 | 5, 9, 12 | pm5.21nd 802 | 1 ⊢ (𝐵 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {copab 5205 × cxp 5683 | 
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 | 
| This theorem is referenced by: (None) | 
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