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Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 14311, having a shorter proof, but requiring that 𝐴 is a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
s1dmALT | ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14301 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | dmeqd 5813 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = dom {〈0, 𝐴〉}) |
3 | dmsnopg 6115 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {〈0, 𝐴〉} = {0}) | |
4 | 2, 3 | eqtrd 2780 | 1 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {csn 4567 〈cop 4573 dom cdm 5590 0cc0 10872 〈“cs1 14298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-iota 6390 df-fun 6434 df-fv 6440 df-s1 14299 |
This theorem is referenced by: (None) |
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