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Mirrors > Home > MPE Home > Th. List > djurcl | Structured version Visualization version GIF version |
Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djurcl | ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3512 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
2 | 1oex 8110 | . . . . 5 ⊢ 1o ∈ V | |
3 | 2 | snid 4601 | . . . 4 ⊢ 1o ∈ {1o} |
4 | opelxpi 5592 | . . . 4 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) | |
5 | 3, 4 | mpan 688 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) |
6 | opeq2 4804 | . . . 4 ⊢ (𝑥 = 𝐶 → 〈1o, 𝑥〉 = 〈1o, 𝐶〉) | |
7 | df-inr 9332 | . . . 4 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | fvmptg 6766 | . . 3 ⊢ ((𝐶 ∈ V ∧ 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) → (inr‘𝐶) = 〈1o, 𝐶〉) |
9 | 1, 5, 8 | syl2anc 586 | . 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = 〈1o, 𝐶〉) |
10 | elun2 4153 | . . . 4 ⊢ (〈1o, 𝐶〉 ∈ ({1o} × 𝐵) → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
11 | 5, 10 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
12 | df-dju 9330 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
13 | 11, 12 | eleqtrrdi 2924 | . 2 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
14 | 9, 13 | eqeltrd 2913 | 1 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∅c0 4291 {csn 4567 〈cop 4573 × cxp 5553 ‘cfv 6355 1oc1o 8095 ⊔ cdju 9327 inrcinr 9329 |
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-pr 5330 ax-un 7461 |
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-rab 3147 df-v 3496 df-sbc 3773 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-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-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 df-1o 8102 df-dju 9330 df-inr 9332 |
This theorem is referenced by: inrresf 9345 updjudhcoinrg 9362 |
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