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Mirrors > Home > MPE Home > Th. List > Mathboxes > setsstrset | Structured version Visualization version GIF version |
Description: Relation between df-sets 16478 and df-strset 34319. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.) |
Ref | Expression |
---|---|
setsstrset | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet 〈(𝐴‘ndx), 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsval 16501 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈(𝐴‘ndx), 𝐵〉) = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {〈(𝐴‘ndx), 𝐵〉})) | |
2 | df-strset 34319 | . 2 ⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {〈(𝐴‘ndx), 𝐵〉}) | |
3 | 1, 2 | syl6reqr 2872 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet 〈(𝐴‘ndx), 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 {csn 4557 〈cop 4563 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 sSet csts 16469 [cstrset 34318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-sets 16478 df-strset 34319 |
This theorem is referenced by: (None) |
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