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Mirrors > Home > ILE Home > Th. List > djur | GIF version |
Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
djur | β’ (πΆ β (π΄ β π΅) β (βπ₯ β π΄ πΆ = (inlβπ₯) β¨ βπ₯ β π΅ πΆ = (inrβπ₯))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldju 7067 | . 2 β’ (πΆ β (π΄ β π΅) β (βπ₯ β π΄ πΆ = ((inl βΎ π΄)βπ₯) β¨ βπ₯ β π΅ πΆ = ((inr βΎ π΅)βπ₯))) | |
2 | fvres 5540 | . . . . 5 β’ (π₯ β π΄ β ((inl βΎ π΄)βπ₯) = (inlβπ₯)) | |
3 | 2 | eqeq2d 2189 | . . . 4 β’ (π₯ β π΄ β (πΆ = ((inl βΎ π΄)βπ₯) β πΆ = (inlβπ₯))) |
4 | 3 | rexbiia 2492 | . . 3 β’ (βπ₯ β π΄ πΆ = ((inl βΎ π΄)βπ₯) β βπ₯ β π΄ πΆ = (inlβπ₯)) |
5 | fvres 5540 | . . . . 5 β’ (π₯ β π΅ β ((inr βΎ π΅)βπ₯) = (inrβπ₯)) | |
6 | 5 | eqeq2d 2189 | . . . 4 β’ (π₯ β π΅ β (πΆ = ((inr βΎ π΅)βπ₯) β πΆ = (inrβπ₯))) |
7 | 6 | rexbiia 2492 | . . 3 β’ (βπ₯ β π΅ πΆ = ((inr βΎ π΅)βπ₯) β βπ₯ β π΅ πΆ = (inrβπ₯)) |
8 | 4, 7 | orbi12i 764 | . 2 β’ ((βπ₯ β π΄ πΆ = ((inl βΎ π΄)βπ₯) β¨ βπ₯ β π΅ πΆ = ((inr βΎ π΅)βπ₯)) β (βπ₯ β π΄ πΆ = (inlβπ₯) β¨ βπ₯ β π΅ πΆ = (inrβπ₯))) |
9 | 1, 8 | bitri 184 | 1 β’ (πΆ β (π΄ β π΅) β (βπ₯ β π΄ πΆ = (inlβπ₯) β¨ βπ₯ β π΅ πΆ = (inrβπ₯))) |
Colors of variables: wff set class |
Syntax hints: β wb 105 β¨ wo 708 = wceq 1353 β wcel 2148 βwrex 2456 βΎ cres 4629 βcfv 5217 β cdju 7036 inlcinl 7044 inrcinr 7045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-1o 6417 df-dju 7037 df-inl 7046 df-inr 7047 |
This theorem is referenced by: djuss 7069 updjud 7081 omp1eomlem 7093 0ct 7106 ctmlemr 7107 ctssdclemn0 7109 fodjuomnilemdc 7142 exmidfodomrlemeldju 7198 |
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