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| Description: All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| str0.a | ⊢ 𝐹 = Slot 𝐼 | 
| Ref | Expression | 
|---|---|
| str0 | ⊢ ∅ = (𝐹‘∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 2 | str0.a | . . 3 ⊢ 𝐹 = Slot 𝐼 | |
| 3 | 1, 2 | strfvn 17223 | . 2 ⊢ (𝐹‘∅) = (∅‘𝐼) | 
| 4 | 0fv 6950 | . 2 ⊢ (∅‘𝐼) = ∅ | |
| 5 | 3, 4 | eqtr2i 2766 | 1 ⊢ ∅ = (𝐹‘∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∅c0 4333 ‘cfv 6561 Slot cslot 17218 | 
| 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-nfc 2892 df-ne 2941 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-mpt 5226 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-slot 17219 | 
| This theorem is referenced by: strfvi 17227 setsnid 17245 setsnidOLD 17246 base0 17252 resseqnbas 17287 resslemOLD 17288 oppchomfval 17757 fuchom 18009 xpchomfval 18224 xpccofval 18227 oduleval 18334 0pos 18367 frmdplusg 18867 efmndplusg 18893 oppgplusfval 19366 mgpplusg 20141 opprmulfval 20336 sralem 21175 sralemOLD 21176 srasca 21183 srascaOLD 21184 sravsca 21185 sravscaOLD 21186 sraip 21187 zlmlem 21527 zlmlemOLD 21528 zlmvsca 21536 thlle 21716 thlleOLD 21717 thloc 21719 psrplusg 21956 psrmulr 21962 psrvscafval 21968 opsrle 22065 ply1plusgfvi 22243 psr1sca2 22252 ply1sca2 22255 resstopn 23194 tnglem 24653 tnglemOLD 24654 tngds 24668 tngdsOLD 24669 ttglem 28885 ttglemOLD 28886 iedgval0 29057 resvlem 33357 resvlemOLD 33358 sn-base0 42505 mendplusgfval 43193 mendmulrfval 43195 mendsca 43197 mendvscafval 43198 | 
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