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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 14636 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
| 2 | 1 | fveq1d 6908 | . 2 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = ({〈0, 𝐴〉}‘0)) | 
| 3 | 0nn0 12541 | . . 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 2777 | 1 ⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ‘cfv 6561 0cc0 11155 ℕ0cn0 12526 〈“cs1 14633 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-n0 12527 df-s1 14634 | 
| This theorem is referenced by: lsws1 14649 eqs1 14650 wrdl1s1 14652 ccats1val2 14665 ccat1st1st 14666 ccat2s1p1 14667 ccat2s1p2 14668 cats1un 14759 revs1 14803 cats1fvn 14897 s2fv0 14926 efgsval2 19751 efgs1 19753 efgsp1 19755 efgsfo 19757 pgpfaclem1 20101 loopclwwlkn1b 30061 clwwlkn1loopb 30062 clwwlknon1 30116 0wlkons1 30140 1wlkdlem4 30159 wlk2v2elem2 30175 ccatws1f1o 32936 cycpmco2lem2 33147 fldext2chn 33769 constrextdg2 33790 signstf0 34583 signsvtn0 34585 signstfvneq0 34587 singoutnword 46897 | 
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