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Mirrors > Home > MPE Home > Th. List > opabbi2dv | Structured version Visualization version GIF version |
Description: Deduce equality of a relation and an ordered-pair class abstraction. Compare abbi2dv 2868. (Contributed by NM, 24-Feb-2014.) |
Ref | Expression |
---|---|
opabbi2dv.1 | ⊢ Rel 𝐴 |
opabbi2dv.3 | ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
opabbi2dv | ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabbi2dv.1 | . . 3 ⊢ Rel 𝐴 | |
2 | opabid2 5785 | . . 3 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴 |
4 | opabbi2dv.3 | . . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓)) | |
5 | 4 | opabbidv 5172 | . 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}) |
6 | 3, 5 | eqtr3id 2787 | 1 ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 {copab 5168 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 df-xp 5640 df-rel 5641 |
This theorem is referenced by: recmulnq 10905 dmscut 27172 dib1dim 39674 |
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