Theorem 1alg 10498

Description: Category 1 has the structure required by Ded and Alg.

Hypothesis

Ref	Expression
1alg.1	|- A e. V

Assertion

Ref	Expression
1alg	|- <.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg

Proof of Theorem 1alg

Step	Hyp	Ref	Expression
1		snex 2740	. . . 4 |- {<.<.A, A>., A>.} e. V
2		snex 2740	. . . 4 |- {<.A, <.A, A>.>.} e. V
3	1, 1, 2	3pm3.2i 816	. . 3 |- ({<.<.A, A>., A>.} e. V /\ {<.<.A, A>., A>.} e. V /\ {<.A, <.A, A>.>.} e. V)
4		snex 2740	. . 3 |- {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} e. V
5		dmsnop 3317	. . . . 5 |- dom {<.<.A, A>., A>.} = {<.A, A>.}
6	5	eqcomi 1471	. . . 4 |- {<.A, A>.} = dom {<.<.A, A>., A>.}
7		dmsnop 3317	. . . . 5 |- dom {<.A, <.A, A>.>.} = {A}
8	7	eqcomi 1471	. . . 4 |- {A} = dom {<.A, <.A, A>.>.}
9	6, 8	isalg 10497	. . 3 |- ((({<.<.A, A>., A>.} e. V /\ {<.<.A, A>., A>.} e. V /\ {<.A, <.A, A>.>.} e. V) /\ {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} e. V) -> (<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg <-> (({<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.A, <.A, A>.>.}:{A}-->{<.A, A>.}) /\ (Fun {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} /\ dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.}) /\ ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ {<.A, A>.}))))
10	3, 4, 9	mp2an 695	. 2 |- (<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg <-> (({<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.A, <.A, A>.>.}:{A}-->{<.A, A>.}) /\ (Fun {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} /\ dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.}) /\ ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ {<.A, A>.})))
11		opex 2772	. . . . 5 |- <.A, A>. e. V
12		1alg.1	. . . . 5 |- A e. V
13	11, 12	f1osn 3704	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
14		f1of 3674	. . . 4 |- ({<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A} -> {<.<.A, A>., A>.}:{<.A, A>.}-->{A})
15	13, 14	ax-mp 7	. . 3 |- {<.<.A, A>., A>.}:{<.A, A>.}-->{A}
16	12, 11	f1osn 3704	. . . 4 |- {<.A, <.A, A>.>.}:{A}-1-1-onto->{<.A, A>.}
17		f1of 3674	. . . 4 |- ({<.A, <.A, A>.>.}:{A}-1-1-onto->{<.A, A>.} -> {<.A, <.A, A>.>.}:{A}-->{<.A, A>.})
18	16, 17	ax-mp 7	. . 3 |- {<.A, <.A, A>.>.}:{A}-->{<.A, A>.}
19	15, 15, 18	3pm3.2i 816	. 2 |- ({<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.A, <.A, A>.>.}:{A}-->{<.A, A>.})
20		opex 2772	. . . 4 |- <.<.A, A>., <.A, A>.>. e. V
21	20, 11	funsn 3529	. . 3 |- Fun {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}
22		dmsnop 3317	. . . 4 |- dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} = {<.<.A, A>., <.A, A>.>.}
23	11, 11	f1osn 3704	. . . . . 6 |- {<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-1-1-onto->{<.A, A>.}
24		f1of 3674	. . . . . 6 |- ({<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-1-1-onto->{<.A, A>.} -> {<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-->{<.A, A>.})
25	23, 24	ax-mp 7	. . . . 5 |- {<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-->{<.A, A>.}
26		fssxp 3622	. . . . 5 |- ({<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-->{<.A, A>.} -> {<.<.A, A>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.}))
27	25, 26	ax-mp 7	. . . 4 |- {<.<.A, A>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.})
28	22, 27	eqsstr 2081	. . 3 |- dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.})
29	20, 11	rnsnop 3436	. . . 4 |- ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} = {<.A, A>.}
30	29	eqimssi 2101	. . 3 |- ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ {<.A, A>.}
31	21, 28, 30	3pm3.2i 816	. 2 |- (Fun {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} /\ dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.}) /\ ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ {<.A, A>.})
32	10, 19, 31	mpbir2an 728	1 |- <.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 773   e. wcel 955  Vcvv 1802   (_ wss 2037  {csn 2399  <.cop 2401   X. cxp 3158  dom cdm 3160  ran crn 3161  Fun wfun 3166  -->wf 3168  -1-1-onto->wf1o 3171  Algcalg 10487

This theorem is referenced by:  1ded 10515

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-alg 10492

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