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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-diagval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 35857 views it as the functionalized identity. See df-bj-diag 35856 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-diagval2 | ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-diagval 35857 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴)) | |
2 | idinxpresid 6037 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
3 | 1, 2 | eqtr4di 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3943 I cid 5566 × cxp 5667 ↾ cres 5671 ‘cfv 6532 Idcdiag2 35855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6484 df-fun 6534 df-fv 6540 df-bj-diag 35856 |
This theorem is referenced by: bj-eldiag 35859 bj-eldiag2 35860 |
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