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Mirrors > Home > ILE Home > Th. List > updjudhcoinlf | Unicode version |
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
updjud.f | |
updjud.g | |
updjudhf.h | ⊔ |
Ref | Expression |
---|---|
updjudhcoinlf | inl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | updjud.f | . . . . 5 | |
2 | updjud.g | . . . . 5 | |
3 | updjudhf.h | . . . . 5 ⊔ | |
4 | 1, 2, 3 | updjudhf 7052 | . . . 4 ⊔ |
5 | ffn 5345 | . . . 4 ⊔ ⊔ | |
6 | 4, 5 | syl 14 | . . 3 ⊔ |
7 | inlresf1 7034 | . . . 4 inl ⊔ | |
8 | f1fn 5403 | . . . 4 inl ⊔ inl | |
9 | 7, 8 | mp1i 10 | . . 3 inl |
10 | f1f 5401 | . . . . 5 inl ⊔ inl ⊔ | |
11 | 7, 10 | ax-mp 5 | . . . 4 inl ⊔ |
12 | frn 5354 | . . . 4 inl ⊔ inl ⊔ | |
13 | 11, 12 | mp1i 10 | . . 3 inl ⊔ |
14 | fnco 5304 | . . 3 ⊔ inl inl ⊔ inl | |
15 | 6, 9, 13, 14 | syl3anc 1233 | . 2 inl |
16 | ffn 5345 | . . 3 | |
17 | 1, 16 | syl 14 | . 2 |
18 | fvco2 5563 | . . . 4 inl inl inl | |
19 | 9, 18 | sylan 281 | . . 3 inl inl |
20 | fvres 5518 | . . . . . 6 inl inl | |
21 | 20 | adantl 275 | . . . . 5 inl inl |
22 | 21 | fveq2d 5498 | . . . 4 inl inl |
23 | 3 | a1i 9 | . . . . 5 ⊔ |
24 | fveq2 5494 | . . . . . . . . 9 inl inl | |
25 | 24 | eqeq1d 2179 | . . . . . . . 8 inl inl |
26 | fveq2 5494 | . . . . . . . . 9 inl inl | |
27 | 26 | fveq2d 5498 | . . . . . . . 8 inl inl |
28 | 26 | fveq2d 5498 | . . . . . . . 8 inl inl |
29 | 25, 27, 28 | ifbieq12d 3551 | . . . . . . 7 inl inl inl inl |
30 | 29 | adantl 275 | . . . . . 6 inl inl inl inl |
31 | 1stinl 7047 | . . . . . . . . 9 inl | |
32 | 31 | adantl 275 | . . . . . . . 8 inl |
33 | 32 | adantr 274 | . . . . . . 7 inl inl |
34 | 33 | iftrued 3532 | . . . . . 6 inl inl inl inl inl |
35 | 30, 34 | eqtrd 2203 | . . . . 5 inl inl |
36 | djulcl 7024 | . . . . . 6 inl ⊔ | |
37 | 36 | adantl 275 | . . . . 5 inl ⊔ |
38 | 1 | adantr 274 | . . . . . 6 |
39 | 2ndinl 7048 | . . . . . . . 8 inl | |
40 | 39 | adantl 275 | . . . . . . 7 inl |
41 | simpr 109 | . . . . . . 7 | |
42 | 40, 41 | eqeltrd 2247 | . . . . . 6 inl |
43 | 38, 42 | ffvelrnd 5629 | . . . . 5 inl |
44 | 23, 35, 37, 43 | fvmptd 5575 | . . . 4 inl inl |
45 | 22, 44 | eqtrd 2203 | . . 3 inl inl |
46 | 40 | fveq2d 5498 | . . 3 inl |
47 | 19, 45, 46 | 3eqtrd 2207 | . 2 inl |
48 | 15, 17, 47 | eqfnfvd 5594 | 1 inl |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wss 3121 c0 3414 cif 3525 cmpt 4048 crn 4610 cres 4611 ccom 4613 wfn 5191 wf 5192 wf1 5193 cfv 5196 c1st 6114 c2nd 6115 ⊔ cdju 7010 inlcinl 7018 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1st 6116 df-2nd 6117 df-1o 6392 df-dju 7011 df-inl 7020 df-inr 7021 |
This theorem is referenced by: updjud 7055 |
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