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Mirrors > Home > ILE Home > Th. List > djurclr | Unicode version |
Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
Ref | Expression |
---|---|
djurclr | inr ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5504 | . 2 inr inr | |
2 | elex 2732 | . . . 4 | |
3 | 1oex 6383 | . . . . . 6 | |
4 | 3 | snid 3601 | . . . . 5 |
5 | opelxpi 4630 | . . . . 5 | |
6 | 4, 5 | mpan 421 | . . . 4 |
7 | opeq2 3753 | . . . . 5 | |
8 | df-inr 7004 | . . . . 5 inr | |
9 | 7, 8 | fvmptg 5556 | . . . 4 inr |
10 | 2, 6, 9 | syl2anc 409 | . . 3 inr |
11 | elun2 3285 | . . . . 5 | |
12 | 6, 11 | syl 14 | . . . 4 |
13 | df-dju 6994 | . . . 4 ⊔ | |
14 | 12, 13 | eleqtrrdi 2258 | . . 3 ⊔ |
15 | 10, 14 | eqeltrd 2241 | . 2 inr ⊔ |
16 | 1, 15 | eqeltrd 2241 | 1 inr ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 cvv 2721 cun 3109 c0 3404 csn 3570 cop 3573 cxp 4596 cres 4600 cfv 5182 c1o 6368 ⊔ cdju 6993 inrcinr 7002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-1o 6375 df-dju 6994 df-inr 7004 |
This theorem is referenced by: inrresf1 7018 |
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