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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 14633 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | fveq1d 6909 | . 2 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = ({〈0, 𝐴〉}‘0)) |
3 | 0nn0 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | fvsng 7200 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → ({〈0, 𝐴〉}‘0) = 𝐴) | |
5 | 3, 4 | mpan 690 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({〈0, 𝐴〉}‘0) = 𝐴) |
6 | 2, 5 | eqtrd 2775 | 1 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 ‘cfv 6563 0cc0 11153 ℕ0cn0 12524 〈“cs1 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-n0 12525 df-s1 14631 |
This theorem is referenced by: lsws1 14646 eqs1 14647 wrdl1s1 14649 ccats1val2 14662 ccat1st1st 14663 ccat2s1p1 14664 ccat2s1p2 14665 cats1un 14756 revs1 14800 cats1fvn 14894 s2fv0 14923 efgsval2 19766 efgs1 19768 efgsp1 19770 efgsfo 19772 pgpfaclem1 20116 loopclwwlkn1b 30071 clwwlkn1loopb 30072 clwwlknon1 30126 0wlkons1 30150 1wlkdlem4 30169 wlk2v2elem2 30185 ccatws1f1o 32921 cycpmco2lem2 33130 fldext2chn 33734 signstf0 34562 signsvtn0 34564 signstfvneq0 34566 singoutnword 46836 |
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