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| Mirrors > Home > MPE Home > Th. List > dmopabelb | Structured version Visualization version GIF version | ||
| Description: A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.) |
| Ref | Expression |
|---|---|
| dmopabel.d | ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| dmopabelb | ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑦𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmopab 5926 | . . 3 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑}) |
| 3 | dmopabel.d | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | exbidv 1921 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓)) |
| 5 | eqid 2737 | . . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
| 6 | 4, 5 | elab2g 3680 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓)) |
| 7 | 2, 6 | bitrid 283 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 {copab 5205 dom cdm 5685 |
| 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 |
| 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-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-br 5144 df-opab 5206 df-dm 5695 |
| This theorem is referenced by: dmopab2rex 5928 dmopab3rexdif 35410 |
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