Theorem 1alg 10654

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 2750	. . . 4 |- {<.<.A, A>., A>.} e. V
2		snex 2750	. . . 4 |- {<.A, <.A, A>.>.} e. V
3	1, 1, 2	3pm3.2i 818	. . 3 |- ({<.<.A, A>., A>.} e. V /\ {<.<.A, A>., A>.} e. V /\ {<.A, <.A, A>.>.} e. V)
4		snex 2750	. . 3 |- {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} e. V
5		dmsnop 3328	. . . . 5 |- dom {<.<.A, A>., A>.} = {<.A, A>.}
6	5	eqcomi 1479	. . . 4 |- {<.A, A>.} = dom {<.<.A, A>., A>.}
7		dmsnop 3328	. . . . 5 |- dom {<.A, <.A, A>.>.} = {A}
8	7	eqcomi 1479	. . . 4 |- {A} = dom {<.A, <.A, A>.>.}
9	6, 8	isalg 10653	. . 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 697	. 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 2782	. . . . 5 |- <.A, A>. e. V
12		1alg.1	. . . . 5 |- A e. V
13	11, 12	f1osn 3719	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
14		f1of 3689	. . . 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 3719	. . . 4 |- {<.A, <.A, A>.>.}:{A}-1-1-onto->{<.A, A>.}
17		f1of 3689	. . . 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 818	. 2 |- ({<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.<.A, A>., A>.}:{<.A, A>.}-->{A} /\ {<.A, <.A, A>.>.}:{A}-->{<.A, A>.})
20		opex 2782	. . . 4 |- <.<.A, A>., <.A, A>.>. e. V
21	20, 11	funsn 3543	. . 3 |- Fun {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}
22		dmsnop 3328	. . . 4 |- dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} = {<.<.A, A>., <.A, A>.>.}
23	11, 11	f1osn 3719	. . . . . 6 |- {<.<.A, A>., <.A, A>.>.}:{<.A, A>.}-1-1-onto->{<.A, A>.}
24		f1of 3689	. . . . . 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 3637	. . . . 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 2091	. . 3 |- dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ ({<.A, A>.} X. {<.A, A>.})
29	20, 11	rnsnop 3450	. . . 4 |- ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} = {<.A, A>.}
30	29	eqimssi 2111	. . 3 |- ran {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} (_ {<.A, A>.}
31	21, 28, 30	3pm3.2i 818	. 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 730	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 775   e. wcel 958  Vcvv 1811   (_ wss 2047  {csn 2409  <.cop 2411   X. cxp 3168  dom cdm 3170  ran crn 3171  Fun wfun 3176  -->wf 3178  -1-1-onto->wf1o 3181  Algcalg 10643

This theorem is referenced by:  1ded 10671

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-alg 10648

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