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Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 14165, 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 14155 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | dmeqd 5774 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = dom {〈0, 𝐴〉}) |
3 | dmsnopg 6076 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {〈0, 𝐴〉} = {0}) | |
4 | 2, 3 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4541 〈cop 4547 dom cdm 5551 0cc0 10729 〈“cs1 14152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-s1 14153 |
This theorem is referenced by: (None) |
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