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Mirrors > Home > ILE Home > Th. List > updjudhcoinrg | Unicode version |
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
updjud.f | |
updjud.g | |
updjudhf.h | ⊔ |
Ref | Expression |
---|---|
updjudhcoinrg | inr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | updjud.f | . . . . 5 | |
2 | updjud.g | . . . . 5 | |
3 | updjudhf.h | . . . . 5 ⊔ | |
4 | 1, 2, 3 | updjudhf 6932 | . . . 4 ⊔ |
5 | ffn 5242 | . . . 4 ⊔ ⊔ | |
6 | 4, 5 | syl 14 | . . 3 ⊔ |
7 | inrresf1 6915 | . . . 4 inr ⊔ | |
8 | f1fn 5300 | . . . 4 inr ⊔ inr | |
9 | 7, 8 | mp1i 10 | . . 3 inr |
10 | f1f 5298 | . . . . 5 inr ⊔ inr ⊔ | |
11 | 7, 10 | ax-mp 5 | . . . 4 inr ⊔ |
12 | frn 5251 | . . . 4 inr ⊔ inr ⊔ | |
13 | 11, 12 | mp1i 10 | . . 3 inr ⊔ |
14 | fnco 5201 | . . 3 ⊔ inr inr ⊔ inr | |
15 | 6, 9, 13, 14 | syl3anc 1201 | . 2 inr |
16 | ffn 5242 | . . 3 | |
17 | 2, 16 | syl 14 | . 2 |
18 | fvco2 5458 | . . . 4 inr inr inr | |
19 | 9, 18 | sylan 281 | . . 3 inr inr |
20 | fvres 5413 | . . . . . 6 inr inr | |
21 | 20 | adantl 275 | . . . . 5 inr inr |
22 | 21 | fveq2d 5393 | . . . 4 inr inr |
23 | 3 | a1i 9 | . . . . 5 ⊔ |
24 | fveq2 5389 | . . . . . . . . 9 inr inr | |
25 | 24 | eqeq1d 2126 | . . . . . . . 8 inr inr |
26 | fveq2 5389 | . . . . . . . . 9 inr inr | |
27 | 26 | fveq2d 5393 | . . . . . . . 8 inr inr |
28 | 26 | fveq2d 5393 | . . . . . . . 8 inr inr |
29 | 25, 27, 28 | ifbieq12d 3468 | . . . . . . 7 inr inr inr inr |
30 | 29 | adantl 275 | . . . . . 6 inr inr inr inr |
31 | 1stinr 6929 | . . . . . . . . . 10 inr | |
32 | 1n0 6297 | . . . . . . . . . . . 12 | |
33 | 32 | neii 2287 | . . . . . . . . . . 11 |
34 | eqeq1 2124 | . . . . . . . . . . 11 inr inr | |
35 | 33, 34 | mtbiri 649 | . . . . . . . . . 10 inr inr |
36 | 31, 35 | syl 14 | . . . . . . . . 9 inr |
37 | 36 | adantl 275 | . . . . . . . 8 inr |
38 | 37 | adantr 274 | . . . . . . 7 inr inr |
39 | 38 | iffalsed 3454 | . . . . . 6 inr inr inr inr inr |
40 | 30, 39 | eqtrd 2150 | . . . . 5 inr inr |
41 | djurcl 6905 | . . . . . 6 inr ⊔ | |
42 | 41 | adantl 275 | . . . . 5 inr ⊔ |
43 | 2 | adantr 274 | . . . . . 6 |
44 | 2ndinr 6930 | . . . . . . . 8 inr | |
45 | 44 | adantl 275 | . . . . . . 7 inr |
46 | simpr 109 | . . . . . . 7 | |
47 | 45, 46 | eqeltrd 2194 | . . . . . 6 inr |
48 | 43, 47 | ffvelrnd 5524 | . . . . 5 inr |
49 | 23, 40, 42, 48 | fvmptd 5470 | . . . 4 inr inr |
50 | 22, 49 | eqtrd 2150 | . . 3 inr inr |
51 | 45 | fveq2d 5393 | . . 3 inr |
52 | 19, 50, 51 | 3eqtrd 2154 | . 2 inr |
53 | 15, 17, 52 | eqfnfvd 5489 | 1 inr |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1316 wcel 1465 wss 3041 c0 3333 cif 3444 cmpt 3959 crn 4510 cres 4511 ccom 4513 wfn 5088 wf 5089 wf1 5090 cfv 5093 c1st 6004 c2nd 6005 c1o 6274 ⊔ cdju 6890 inrcinr 6899 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1st 6006 df-2nd 6007 df-1o 6281 df-dju 6891 df-inl 6900 df-inr 6901 |
This theorem is referenced by: updjud 6935 |
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