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Mirrors > Home > MPE Home > Th. List > dmopabss | Structured version Visualization version GIF version |
Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.) |
Ref | Expression |
---|---|
dmopabss | ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmopab 5748 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 19.42v 1954 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)) | |
3 | 2 | abbii 2863 | . . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} |
4 | ssab2 4006 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴 | |
5 | 3, 4 | eqsstri 3949 | . 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
6 | 1, 5 | eqsstri 3949 | 1 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1781 ∈ wcel 2111 {cab 2776 ⊆ wss 3881 {copab 5092 dom cdm 5519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-dm 5529 |
This theorem is referenced by: fvopab4ndm 6774 opabex 6960 perpln1 26504 dmadjss 29670 abrexdomjm 30275 abrexdom 35168 |
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