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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 9941 | . . 3 β’ inl:Vβ1-1-ontoβ({β } Γ V) | |
2 | f1ofun 6844 | . . 3 β’ (inl:Vβ1-1-ontoβ({β } Γ V) β Fun inl) | |
3 | ffvresb 7138 | . . 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 3490 | . . . 4 β’ (π₯ β π΄ β π₯ β V) | |
6 | opex 5468 | . . . . 5 β’ β¨β , π₯β© β V | |
7 | df-inl 9931 | . . . . 5 β’ inl = (π₯ β V β¦ β¨β , π₯β©) | |
8 | 6, 7 | dmmpti 6702 | . . . 4 β’ dom inl = V |
9 | 5, 8 | eleqtrrdi 2839 | . . 3 β’ (π₯ β π΄ β π₯ β dom inl) |
10 | djulcl 9939 | . . 3 β’ (π₯ β π΄ β (inlβπ₯) β (π΄ β π΅)) | |
11 | 9, 10 | jca 510 | . 2 β’ (π₯ β π΄ β (π₯ β dom inl β§ (inlβπ₯) β (π΄ β π΅))) |
12 | 4, 11 | mprgbir 3064 | 1 β’ (inl βΎ π΄):π΄βΆ(π΄ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 β wcel 2098 βwral 3057 Vcvv 3471 β c0 4324 {csn 4630 β¨cop 4636 Γ cxp 5678 dom cdm 5680 βΎ cres 5682 Fun wfun 6545 βΆwf 6547 β1-1-ontoβwf1o 6550 βcfv 6551 β cdju 9927 inlcinl 9928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-1st 7997 df-2nd 7998 df-dju 9930 df-inl 9931 |
This theorem is referenced by: inlresf1 9944 updjudhcoinlf 9961 updjud 9963 |
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