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Mirrors > Home > MPE Home > Th. List > issubc2 | Structured version Visualization version GIF version |
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
issubc.h | β’ π» = (Homf βπΆ) |
issubc.i | β’ 1 = (IdβπΆ) |
issubc.o | β’ Β· = (compβπΆ) |
issubc.c | β’ (π β πΆ β Cat) |
issubc2.a | β’ (π β π½ Fn (π Γ π)) |
Ref | Expression |
---|---|
issubc2 | β’ (π β (π½ β (SubcatβπΆ) β (π½ βcat π» β§ βπ₯ β π (( 1 βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubc.h | . 2 β’ π» = (Homf βπΆ) | |
2 | issubc.i | . 2 β’ 1 = (IdβπΆ) | |
3 | issubc.o | . 2 β’ Β· = (compβπΆ) | |
4 | issubc.c | . 2 β’ (π β πΆ β Cat) | |
5 | issubc2.a | . . . . 5 β’ (π β π½ Fn (π Γ π)) | |
6 | 5 | fndmd 6647 | . . . 4 β’ (π β dom π½ = (π Γ π)) |
7 | 6 | dmeqd 5898 | . . 3 β’ (π β dom dom π½ = dom (π Γ π)) |
8 | dmxpid 5922 | . . 3 β’ dom (π Γ π) = π | |
9 | 7, 8 | eqtr2di 2783 | . 2 β’ (π β π = dom dom π½) |
10 | 1, 2, 3, 4, 9 | issubc 17791 | 1 β’ (π β (π½ β (SubcatβπΆ) β (π½ βcat π» β§ βπ₯ β π (( 1 βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 β¨cop 4629 class class class wbr 5141 Γ cxp 5667 dom cdm 5669 Fn wfn 6531 βcfv 6536 (class class class)co 7404 compcco 17215 Catccat 17614 Idccid 17615 Homf chomf 17616 βcat cssc 17760 Subcatcsubc 17762 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 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-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 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-ima 5682 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-ov 7407 df-oprab 7408 df-mpo 7409 df-pm 8822 df-ixp 8891 df-ssc 17763 df-subc 17765 |
This theorem is referenced by: 0subcat 17794 catsubcat 17795 subcidcl 17800 subccocl 17801 issubc3 17805 fullsubc 17806 rnghmsubcsetc 20526 rhmsubcsetc 20555 rhmsubcrngc 20561 srhmsubc 20573 rhmsubc 20582 rhmsubcALTV 47217 srhmsubcALTV 47257 |
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