Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 13388, 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 13378 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | dmeqd 5326 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = dom {〈0, 𝐴〉}) |
3 | dmsnopg 5606 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {〈0, 𝐴〉} = {0}) | |
4 | 2, 3 | eqtrd 2656 | 1 ⊢ (𝐴 ∈ 𝑆 → dom 〈“𝐴”〉 = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {csn 4177 〈cop 4183 dom cdm 5114 0cc0 9936 〈“cs1 13294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-s1 13302 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |