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Mirrors > Home > ILE Home > Th. List > Mathboxes > djurclALT | GIF version |
Description: Shortening of djurcl 6852 using djucllem 12588. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
djurclALT | ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 6251 | . . . . 5 ⊢ 1o ∈ V | |
2 | df-inr 6848 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
3 | 1, 2 | djucllem 12588 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵)) |
4 | 3 | olcd 694 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (((inr ↾ 𝐵)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵))) |
5 | elun 3164 | . . 3 ⊢ (((inr ↾ 𝐵)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (((inr ↾ 𝐵)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵))) | |
6 | 4, 5 | sylibr 133 | . 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
7 | df-dju 6838 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
8 | 6, 7 | syl6eleqr 2193 | 1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 670 ∈ wcel 1448 ∪ cun 3019 ∅c0 3310 {csn 3474 × cxp 4475 ↾ cres 4479 ‘cfv 5059 1oc1o 6236 ⊔ cdju 6837 inrcinr 6846 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-res 4489 df-iota 5024 df-fun 5061 df-fv 5067 df-1o 6243 df-dju 6838 df-inr 6848 |
This theorem is referenced by: (None) |
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