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Mirrors > Home > MPE Home > Th. List > inlresf | Structured version Visualization version GIF version |
Description: The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
inlresf | β’ (inl βΎ π΄):π΄βΆ(π΄ β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1o 9856 | . . 3 β’ inl:Vβ1-1-ontoβ({β } Γ V) | |
2 | f1ofun 6790 | . . 3 β’ (inl:Vβ1-1-ontoβ({β } Γ V) β Fun inl) | |
3 | ffvresb 7076 | . . 3 β’ (Fun inl β ((inl βΎ π΄):π΄βΆ(π΄ β π΅) β βπ₯ β π΄ (π₯ β dom inl β§ (inlβπ₯) β (π΄ β π΅)))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 β’ ((inl βΎ π΄):π΄βΆ(π΄ β π΅) β βπ₯ β π΄ (π₯ β dom inl β§ (inlβπ₯) β (π΄ β π΅))) |
5 | elex 3465 | . . . 4 β’ (π₯ β π΄ β π₯ β V) | |
6 | opex 5425 | . . . . 5 β’ β¨β , π₯β© β V | |
7 | df-inl 9846 | . . . . 5 β’ inl = (π₯ β V β¦ β¨β , π₯β©) | |
8 | 6, 7 | dmmpti 6649 | . . . 4 β’ dom inl = V |
9 | 5, 8 | eleqtrrdi 2845 | . . 3 β’ (π₯ β π΄ β π₯ β dom inl) |
10 | djulcl 9854 | . . 3 β’ (π₯ β π΄ β (inlβπ₯) β (π΄ β π΅)) | |
11 | 9, 10 | jca 513 | . 2 β’ (π₯ β π΄ β (π₯ β dom inl β§ (inlβπ₯) β (π΄ β π΅))) |
12 | 4, 11 | mprgbir 3068 | 1 β’ (inl βΎ π΄):π΄βΆ(π΄ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β wcel 2107 βwral 3061 Vcvv 3447 β c0 4286 {csn 4590 β¨cop 4596 Γ cxp 5635 dom cdm 5637 βΎ cres 5639 Fun wfun 6494 βΆwf 6496 β1-1-ontoβwf1o 6499 βcfv 6500 β cdju 9842 inlcinl 9843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-1st 7925 df-2nd 7926 df-dju 9845 df-inl 9846 |
This theorem is referenced by: inlresf1 9859 updjudhcoinlf 9876 updjud 9878 |
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