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Mathbox for Jeff Madsen |
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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 5758 | . . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V)) | |
2 | opelxp1 5711 | . . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V) |
4 | 3 | anim1i 614 | . . 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 3489 | . . . 4 ⊢ (𝜒 → 𝐴 ∈ V) |
8 | 7 | anim1i 614 | . . 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 5531 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)) |
13 | 5, 9, 12 | pm5.21nd 799 | 1 ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 {copab 5203 × cxp 5667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-xp 5675 |
This theorem is referenced by: (None) |
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