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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 5784 | . . 3 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑}) |
3 | dmopabel.d | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
4 | 3 | exbidv 1922 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓)) |
5 | eqid 2821 | . . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑} | |
6 | 4, 5 | elab2g 3668 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓)) |
7 | 2, 6 | syl5bb 285 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 {copab 5128 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-dm 5565 |
This theorem is referenced by: dmopab2rex 5786 dmopab3rexdif 32652 |
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