![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5929 | . . 3 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑}) |
3 | dmopabel.d | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
4 | 3 | exbidv 1919 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓)) |
5 | eqid 2735 | . . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
6 | 4, 5 | elab2g 3683 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 {copab 5210 dom cdm 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-dm 5699 |
This theorem is referenced by: dmopab2rex 5931 dmopab3rexdif 35390 |
Copyright terms: Public domain | W3C validator |