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| Mirrors > Home > MPE Home > Th. List > strfvn | Structured version Visualization version GIF version | ||
| Description: Value of a structure
component extractor 𝐸.  Normally, 𝐸 is a
       defined constant symbol such as Base (df-base 17248) and 𝑁 is the
       index of the component. 𝑆 is a structure, i.e. a specific
member of
       a class of structures such as Poset (df-poset 18359) where
       𝑆
∈ Poset. Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. Alternatively, use strfv 17240 instead of strfvn 17223. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| strfvn.f | ⊢ 𝑆 ∈ V | 
| strfvn.c | ⊢ 𝐸 = Slot 𝑁 | 
| Ref | Expression | 
|---|---|
| strfvn | ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | strfvn.c | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
| 2 | strfvn.f | . . . 4 ⊢ 𝑆 ∈ V | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → 𝑆 ∈ V) | 
| 4 | 1, 3 | strfvnd 17222 | . 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁)) | 
| 5 | 4 | mptru 1547 | 1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 ‘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: str0 17226 ndxarg 17233 setsnid 17245 setsnidOLD 17246 baseval 17249 ressbasOLD 17281 resvsca 33356 | 
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