Theorem besubbeca 15196

Description: Lemma to simplify some subcategories related theorems .

Assertion

Ref	Expression
besubbeca	|- (U e. ( SubCat ` T) -> T e. Cat )

Proof of Theorem besubbeca

Step	Hyp	Ref	Expression
1		elfvdm 4704	. 2 |- (U e. ( SubCat ` T) -> T e. dom SubCat )
2		df-subc 15192	. . . . 5 |- SubCat = {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})}
3	2	dmeqi 4158	. . . 4 |- dom SubCat = dom {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})}
4	3	eleq2i 1961	. . 3 |- (T e. dom SubCat <-> T e. dom {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})})
5		dmopabss 4168	. . . 4 |- dom {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})} C_ Cat
6	5	sseli 2617	. . 3 |- (T e. dom {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})} -> T e. Cat )
7	4, 6	sylbi 216	. 2 |- (T e. dom SubCat -> T e. Cat )
8	1, 7	syl 12	1 |- (U e. ( SubCat ` T) -> T e. Cat )

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {crab 2108   C_ wss 2593  {copab 3395  dom cdm 3986  ` cfv 3998  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Cat ccat 15099   SubCat csubc 15191

This theorem is referenced by:  obsubc2 15198  idsubc 15199  domsubc 15200  codsubc 15201  subctct 15202  morsubc 15203  cmpsubc 15204  idsubidsup 15205  idsubfun 15206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-subc 15192

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