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Mirrors > Home > ILE Home > Th. List > updjudhf | Unicode version |
Description: The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
updjud.f | |
updjud.g | |
updjudhf.h | ⊔ |
Ref | Expression |
---|---|
updjudhf | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldju2ndl 6957 | . . . . . 6 ⊔ | |
2 | 1 | ex 114 | . . . . 5 ⊔ |
3 | updjud.f | . . . . . 6 | |
4 | ffvelrn 5553 | . . . . . . 7 | |
5 | 4 | ex 114 | . . . . . 6 |
6 | 3, 5 | syl 14 | . . . . 5 |
7 | 2, 6 | sylan9r 407 | . . . 4 ⊔ |
8 | 7 | imp 123 | . . 3 ⊔ |
9 | df-ne 2309 | . . . . 5 | |
10 | eldju2ndr 6958 | . . . . . . 7 ⊔ | |
11 | 10 | ex 114 | . . . . . 6 ⊔ |
12 | updjud.g | . . . . . . 7 | |
13 | ffvelrn 5553 | . . . . . . . 8 | |
14 | 13 | ex 114 | . . . . . . 7 |
15 | 12, 14 | syl 14 | . . . . . 6 |
16 | 11, 15 | sylan9r 407 | . . . . 5 ⊔ |
17 | 9, 16 | syl5bir 152 | . . . 4 ⊔ |
18 | 17 | imp 123 | . . 3 ⊔ |
19 | eldju1st 6956 | . . . . . 6 ⊔ | |
20 | 1n0 6329 | . . . . . . . 8 | |
21 | neeq1 2321 | . . . . . . . 8 | |
22 | 20, 21 | mpbiri 167 | . . . . . . 7 |
23 | 22 | orim2i 750 | . . . . . 6 |
24 | 19, 23 | syl 14 | . . . . 5 ⊔ |
25 | 24 | adantl 275 | . . . 4 ⊔ |
26 | dcne 2319 | . . . 4 DECID | |
27 | 25, 26 | sylibr 133 | . . 3 ⊔ DECID |
28 | 8, 18, 27 | ifcldadc 3501 | . 2 ⊔ |
29 | updjudhf.h | . 2 ⊔ | |
30 | 28, 29 | fmptd 5574 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wne 2308 c0 3363 cif 3474 cmpt 3989 wf 5119 cfv 5123 c1st 6036 c2nd 6037 c1o 6306 ⊔ cdju 6922 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: updjudhcoinlf 6965 updjudhcoinrg 6966 updjud 6967 |
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