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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 6664 | . . . 4 β’ (π β dom π½ = (π Γ π)) |
7 | 6 | dmeqd 5912 | . . 3 β’ (π β dom dom π½ = dom (π Γ π)) |
8 | dmxpid 5936 | . . 3 β’ dom (π Γ π) = π | |
9 | 7, 8 | eqtr2di 2785 | . 2 β’ (π β π = dom dom π½) |
10 | 1, 2, 3, 4, 9 | issubc 17828 | 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 394 = wceq 1533 β wcel 2098 βwral 3058 β¨cop 4638 class class class wbr 5152 Γ cxp 5680 dom cdm 5682 Fn wfn 6548 βcfv 6553 (class class class)co 7426 compcco 17252 Catccat 17651 Idccid 17652 Homf chomf 17653 βcat cssc 17797 Subcatcsubc 17799 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pm 8854 df-ixp 8923 df-ssc 17800 df-subc 17802 |
This theorem is referenced by: 0subcat 17831 catsubcat 17832 subcidcl 17837 subccocl 17838 issubc3 17842 fullsubc 17843 rnghmsubcsetc 20573 rhmsubcsetc 20602 rhmsubcrngc 20608 srhmsubc 20620 rhmsubc 20629 rhmsubcALTV 47425 srhmsubcALTV 47465 |
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