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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 9906 | . . 3 β’ inl:Vβ1-1-ontoβ({β } Γ V) | |
2 | f1ofun 6828 | . . 3 β’ (inl:Vβ1-1-ontoβ({β } Γ V) β Fun inl) | |
3 | ffvresb 7119 | . . 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 3487 | . . . 4 β’ (π₯ β π΄ β π₯ β V) | |
6 | opex 5457 | . . . . 5 β’ β¨β , π₯β© β V | |
7 | df-inl 9896 | . . . . 5 β’ inl = (π₯ β V β¦ β¨β , π₯β©) | |
8 | 6, 7 | dmmpti 6687 | . . . 4 β’ dom inl = V |
9 | 5, 8 | eleqtrrdi 2838 | . . 3 β’ (π₯ β π΄ β π₯ β dom inl) |
10 | djulcl 9904 | . . 3 β’ (π₯ β π΄ β (inlβπ₯) β (π΄ β π΅)) | |
11 | 9, 10 | jca 511 | . 2 β’ (π₯ β π΄ β (π₯ β dom inl β§ (inlβπ₯) β (π΄ β π΅))) |
12 | 4, 11 | mprgbir 3062 | 1 β’ (inl βΎ π΄):π΄βΆ(π΄ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 β wcel 2098 βwral 3055 Vcvv 3468 β c0 4317 {csn 4623 β¨cop 4629 Γ cxp 5667 dom cdm 5669 βΎ cres 5671 Fun wfun 6530 βΆwf 6532 β1-1-ontoβwf1o 6535 βcfv 6536 β cdju 9892 inlcinl 9893 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-1st 7971 df-2nd 7972 df-dju 9895 df-inl 9896 |
This theorem is referenced by: inlresf1 9909 updjudhcoinlf 9926 updjud 9928 |
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