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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 13658 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | fveq1d 6435 | . 2 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = ({〈0, 𝐴〉}‘0)) |
3 | 0nn0 11635 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | fvsng 6698 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → ({〈0, 𝐴〉}‘0) = 𝐴) | |
5 | 3, 4 | mpan 683 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({〈0, 𝐴〉}‘0) = 𝐴) |
6 | 2, 5 | eqtrd 2861 | 1 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {csn 4397 〈cop 4403 ‘cfv 6123 0cc0 10252 ℕ0cn0 11618 〈“cs1 13655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-mulcl 10314 ax-i2m1 10320 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-n0 11619 df-s1 13656 |
This theorem is referenced by: lsws1 13671 eqs1 13672 wrdl1s1 13674 ccats1val2 13687 ccat1st1st 13688 ccat2s1p1 13689 ccat2s1p2 13690 cats1un 13811 revs1 13881 cats1fvn 13979 s2fv0 14008 efgsval2 18497 efgs1 18499 efgsp1 18501 efgsfo 18504 pgpfaclem1 18834 loopclwwlkn1b 27387 clwwlkn1loopb 27388 clwwlknon1 27471 0wlkons1 27497 1wlkdlem4 27516 wlk2v2elem2 27532 signstf0 31192 signstfvn 31193 signsvtn0 31194 signsvtn0OLD 31195 signstfvneq0 31197 |
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