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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-diagval | Structured version Visualization version GIF version |
Description: Value of the funtionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 34469 views it as the diagonal function. See df-bj-diag 34467 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-diagval | ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-diag 34467 | . 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥)) | |
2 | reseq2 5850 | . 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴)) | |
3 | elex 3514 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | resiexg 7621 | . 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) | |
5 | 1, 2, 3, 4 | fvmptd3 6793 | 1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 I cid 5461 ↾ cres 5559 ‘cfv 6357 Idcdiag2 34466 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-bj-diag 34467 |
This theorem is referenced by: bj-diagval2 34469 |
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