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Mirrors > Home > MPE Home > Th. List > strndxid | Structured version Visualization version GIF version |
Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
Ref | Expression |
---|---|
strndxid.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strndxid.e | ⊢ 𝐸 = Slot 𝑁 |
strndxid.n | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
strndxid | ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strndxid.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | strndxid.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16509 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | strndxid.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 3, 4 | strfvnd 16502 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
6 | 5 | eqcomd 2827 | 1 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 ℕcn 11638 ndxcnx 16480 Slot cslot 16482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-ndx 16486 df-slot 16487 |
This theorem is referenced by: estrreslem1 17387 edgfndxid 26778 bj-endbase 34600 bj-endcomp 34601 |
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