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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 6964 | . . . 4 ⊔ |
5 | ffn 5272 | . . . 4 ⊔ ⊔ | |
6 | 4, 5 | syl 14 | . . 3 ⊔ |
7 | inrresf1 6947 | . . . 4 inr ⊔ | |
8 | f1fn 5330 | . . . 4 inr ⊔ inr | |
9 | 7, 8 | mp1i 10 | . . 3 inr |
10 | f1f 5328 | . . . . 5 inr ⊔ inr ⊔ | |
11 | 7, 10 | ax-mp 5 | . . . 4 inr ⊔ |
12 | frn 5281 | . . . 4 inr ⊔ inr ⊔ | |
13 | 11, 12 | mp1i 10 | . . 3 inr ⊔ |
14 | fnco 5231 | . . 3 ⊔ inr inr ⊔ inr | |
15 | 6, 9, 13, 14 | syl3anc 1216 | . 2 inr |
16 | ffn 5272 | . . 3 | |
17 | 2, 16 | syl 14 | . 2 |
18 | fvco2 5490 | . . . 4 inr inr inr | |
19 | 9, 18 | sylan 281 | . . 3 inr inr |
20 | fvres 5445 | . . . . . 6 inr inr | |
21 | 20 | adantl 275 | . . . . 5 inr inr |
22 | 21 | fveq2d 5425 | . . . 4 inr inr |
23 | 3 | a1i 9 | . . . . 5 ⊔ |
24 | fveq2 5421 | . . . . . . . . 9 inr inr | |
25 | 24 | eqeq1d 2148 | . . . . . . . 8 inr inr |
26 | fveq2 5421 | . . . . . . . . 9 inr inr | |
27 | 26 | fveq2d 5425 | . . . . . . . 8 inr inr |
28 | 26 | fveq2d 5425 | . . . . . . . 8 inr inr |
29 | 25, 27, 28 | ifbieq12d 3498 | . . . . . . 7 inr inr inr inr |
30 | 29 | adantl 275 | . . . . . 6 inr inr inr inr |
31 | 1stinr 6961 | . . . . . . . . . 10 inr | |
32 | 1n0 6329 | . . . . . . . . . . . 12 | |
33 | 32 | neii 2310 | . . . . . . . . . . 11 |
34 | eqeq1 2146 | . . . . . . . . . . 11 inr inr | |
35 | 33, 34 | mtbiri 664 | . . . . . . . . . 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 3484 | . . . . . 6 inr inr inr inr inr |
40 | 30, 39 | eqtrd 2172 | . . . . 5 inr inr |
41 | djurcl 6937 | . . . . . 6 inr ⊔ | |
42 | 41 | adantl 275 | . . . . 5 inr ⊔ |
43 | 2 | adantr 274 | . . . . . 6 |
44 | 2ndinr 6962 | . . . . . . . 8 inr | |
45 | 44 | adantl 275 | . . . . . . 7 inr |
46 | simpr 109 | . . . . . . 7 | |
47 | 45, 46 | eqeltrd 2216 | . . . . . 6 inr |
48 | 43, 47 | ffvelrnd 5556 | . . . . 5 inr |
49 | 23, 40, 42, 48 | fvmptd 5502 | . . . 4 inr inr |
50 | 22, 49 | eqtrd 2172 | . . 3 inr inr |
51 | 45 | fveq2d 5425 | . . 3 inr |
52 | 19, 50, 51 | 3eqtrd 2176 | . 2 inr |
53 | 15, 17, 52 | eqfnfvd 5521 | 1 inr |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 wss 3071 c0 3363 cif 3474 cmpt 3989 crn 4540 cres 4541 ccom 4543 wfn 5118 wf 5119 wf1 5120 cfv 5123 c1st 6036 c2nd 6037 c1o 6306 ⊔ cdju 6922 inrcinr 6931 |
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: updjud 6967 |
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