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Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 13241, having a shorter proof, but requiring that 𝐴 ia 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 13231 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | dmeqd 5248 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = dom {〈0, 𝐴〉}) |
3 | dmsnopg 5524 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {〈0, 𝐴〉} = {0}) | |
4 | 2, 3 | eqtrd 2644 | 1 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 dom cdm 5038 0cc0 9815 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-s1 13157 |
This theorem is referenced by: (None) |
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