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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 14493 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | fveq1d 6849 | . 2 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = ({⟨0, 𝐴⟩}‘0)) |
3 | 0nn0 12435 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | fvsng 7131 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → ({⟨0, 𝐴⟩}‘0) = 𝐴) | |
5 | 3, 4 | mpan 689 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({⟨0, 𝐴⟩}‘0) = 𝐴) |
6 | 2, 5 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4591 ⟨cop 4597 ‘cfv 6501 0cc0 11058 ℕ0cn0 12420 ⟨“cs1 14490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-i2m1 11126 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-n0 12421 df-s1 14491 |
This theorem is referenced by: lsws1 14506 eqs1 14507 wrdl1s1 14509 ccats1val2 14522 ccat1st1st 14523 ccat2s1p1 14524 ccat2s1p2 14525 cats1un 14616 revs1 14660 cats1fvn 14754 s2fv0 14783 efgsval2 19522 efgs1 19524 efgsp1 19526 efgsfo 19528 pgpfaclem1 19867 loopclwwlkn1b 29028 clwwlkn1loopb 29029 clwwlknon1 29083 0wlkons1 29107 1wlkdlem4 29126 wlk2v2elem2 29142 cycpmco2lem2 32018 signstf0 33220 signsvtn0 33222 signstfvneq0 33224 singoutnword 45195 |
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