![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s1fv | Structured version Visualization version GIF version |
Description: Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1fv | ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14544 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | fveq1d 6890 | . 2 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = ({⟨0, 𝐴⟩}‘0)) |
3 | 0nn0 12483 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | fvsng 7174 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → ({⟨0, 𝐴⟩}‘0) = 𝐴) | |
5 | 3, 4 | mpan 688 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({⟨0, 𝐴⟩}‘0) = 𝐴) |
6 | 2, 5 | eqtrd 2772 | 1 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4627 ⟨cop 4633 ‘cfv 6540 0cc0 11106 ℕ0cn0 12468 ⟨“cs1 14541 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-mulcl 11168 ax-i2m1 11174 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-n0 12469 df-s1 14542 |
This theorem is referenced by: lsws1 14557 eqs1 14558 wrdl1s1 14560 ccats1val2 14573 ccat1st1st 14574 ccat2s1p1 14575 ccat2s1p2 14576 cats1un 14667 revs1 14711 cats1fvn 14805 s2fv0 14834 efgsval2 19595 efgs1 19597 efgsp1 19599 efgsfo 19601 pgpfaclem1 19945 loopclwwlkn1b 29284 clwwlkn1loopb 29285 clwwlknon1 29339 0wlkons1 29363 1wlkdlem4 29382 wlk2v2elem2 29398 cycpmco2lem2 32273 signstf0 33567 signsvtn0 33569 signstfvneq0 33571 singoutnword 45582 |
Copyright terms: Public domain | W3C validator |