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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 14554 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | fveq1d 6894 | . 2 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = ({⟨0, 𝐴⟩}‘0)) |
3 | 0nn0 12493 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | fvsng 7181 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → ({⟨0, 𝐴⟩}‘0) = 𝐴) | |
5 | 3, 4 | mpan 686 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({⟨0, 𝐴⟩}‘0) = 𝐴) |
6 | 2, 5 | eqtrd 2770 | 1 ⊢ (𝐴 ∈ 𝐵 → (⟨“𝐴”⟩‘0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 {csn 4629 ⟨cop 4635 ‘cfv 6544 0cc0 11114 ℕ0cn0 12478 ⟨“cs1 14551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-mulcl 11176 ax-i2m1 11182 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-n0 12479 df-s1 14552 |
This theorem is referenced by: lsws1 14567 eqs1 14568 wrdl1s1 14570 ccats1val2 14583 ccat1st1st 14584 ccat2s1p1 14585 ccat2s1p2 14586 cats1un 14677 revs1 14721 cats1fvn 14815 s2fv0 14844 efgsval2 19644 efgs1 19646 efgsp1 19648 efgsfo 19650 pgpfaclem1 19994 loopclwwlkn1b 29560 clwwlkn1loopb 29561 clwwlknon1 29615 0wlkons1 29639 1wlkdlem4 29658 wlk2v2elem2 29674 cycpmco2lem2 32554 signstf0 33875 signsvtn0 33877 signstfvneq0 33879 singoutnword 45896 |
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