Theorem ttcn 14913

Description: A special case of txcnopab 10228, when one function is constant. Bourbaki TG I.26 prop. 4.

Hypotheses

Ref	Expression
ttcnlem.1	|- X = U.J
ttcnlem.2	|- F = {<.x, y>. | (x e. X /\ y = <.x, A>.)}

Assertion

Ref	Expression
ttcn	|- ((J e. Top /\ A e. X) -> F e. (J Cn (J X.t J)))

Distinct variable groups:   x,A,y   x,J,y   x,X,y

Proof of Theorem ttcn

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . 6 |- (a = A -> (a e. X <-> A e. X))
2	1	anbi2d 678	. . . . 5 |- (a = A -> ((J e. Top /\ a e. X) <-> (J e. Top /\ A e. X)))
3		opeq2 3159	. . . . . . . . 9 |- (a = A -> <.x, a>. = <.x, A>.)
4	3	eqeq2d 1895	. . . . . . . 8 |- (a = A -> (y = <.x, a>. <-> y = <.x, A>.))
5	4	anbi2d 678	. . . . . . 7 |- (a = A -> ((x e. X /\ y = <.x, a>.) <-> (x e. X /\ y = <.x, A>.)))
6	5	opabbidv 3401	. . . . . 6 |- (a = A -> {<.x, y>. | (x e. X /\ y = <.x, a>.)} = {<.x, y>. | (x e. X /\ y = <.x, A>.)})
7	6	eleq1d 1963	. . . . 5 |- (a = A -> ({<.x, y>. | (x e. X /\ y = <.x, a>.)} e. (J Cn (J X.t J)) <-> {<.x, y>. | (x e. X /\ y = <.x, A>.)} e. (J Cn (J X.t J))))
8	2, 7	imbi12d 688	. . . 4 |- (a = A -> (((J e. Top /\ a e. X) -> {<.x, y>. | (x e. X /\ y = <.x, a>.)} e. (J Cn (J X.t J))) <-> ((J e. Top /\ A e. X) -> {<.x, y>. | (x e. X /\ y = <.x, A>.)} e. (J Cn (J X.t J)))))
9		simpl 346	. . . . 5 |- ((J e. Top /\ a e. X) -> J e. Top)
10		ttcnlem.1	. . . . . . 7 |- X = U.J
11	10	idcn 9042	. . . . . 6 |- (J e. Top -> ( _I |` X) e. (J Cn J))
12	11	adantr 425	. . . . 5 |- ((J e. Top /\ a e. X) -> ( _I |` X) e. (J Cn J))
13		simpr 350	. . . . . . 7 |- ((J e. Top /\ a e. X) -> a e. X)
14		visset 2295	. . . . . . . 8 |- a e. _V
15	14	fconst 4602	. . . . . . 7 |- (X X. {a}):X-->{a}
16	13, 15	jctir 317	. . . . . 6 |- ((J e. Top /\ a e. X) -> (a e. X /\ (X X. {a}):X-->{a}))
17	10, 10	cnconst 9057	. . . . . 6 |- (((J e. Top /\ J e. Top) /\ (a e. X /\ (X X. {a}):X-->{a})) -> (X X. {a}) e. (J Cn J))
18	9, 9, 16, 17	syl21anc 1099	. . . . 5 |- ((J e. Top /\ a e. X) -> (X X. {a}) e. (J Cn J))
19		eqid 1884	. . . . . 6 |- (J X.t J) = (J X.t J)
20		fvresi 4819	. . . . . . . . . . 11 |- (x e. X -> (( _I |` X)` x) = x)
21	20	eqcomd 1889	. . . . . . . . . 10 |- (x e. X -> x = (( _I |` X)` x))
22	14	fvconst2 4822	. . . . . . . . . . 11 |- (x e. X -> ((X X. {a})` x) = a)
23	22	eqcomd 1889	. . . . . . . . . 10 |- (x e. X -> a = ((X X. {a})` x))
24	21, 23	opeq12d 3166	. . . . . . . . 9 |- (x e. X -> <.x, a>. = <.(( _I |` X)` x), ((X X. {a})` x)>.)
25	24	eqeq2d 1895	. . . . . . . 8 |- (x e. X -> (y = <.x, a>. <-> y = <.(( _I |` X)` x), ((X X. {a})` x)>.))
26	25	pm5.32i 707	. . . . . . 7 |- ((x e. X /\ y = <.x, a>.) <-> (x e. X /\ y = <.(( _I |` X)` x), ((X X. {a})` x)>.))
27	26	opabbii 3402	. . . . . 6 |- {<.x, y>. | (x e. X /\ y = <.x, a>.)} = {<.x, y>. | (x e. X /\ y = <.(( _I |` X)` x), ((X X. {a})` x)>.)}
28	19, 10, 27	txcnopab 10228	. . . . 5 |- (((J e. Top /\ J e. Top /\ J e. Top) /\ (( _I |` X) e. (J Cn J) /\ (X X. {a}) e. (J Cn J))) -> {<.x, y>. | (x e. X /\ y = <.x, a>.)} e. (J Cn (J X.t J)))
29	9, 9, 9, 12, 18, 28	syl32anc 1108	. . . 4 |- ((J e. Top /\ a e. X) -> {<.x, y>. | (x e. X /\ y = <.x, a>.)} e. (J Cn (J X.t J)))
30	8, 29	vtoclg 2346	. . 3 |- (A e. X -> ((J e. Top /\ A e. X) -> {<.x, y>. | (x e. X /\ y = <.x, A>.)} e. (J Cn (J X.t J))))
31	30	anabsi7 555	. 2 |- ((J e. Top /\ A e. X) -> {<.x, y>. | (x e. X /\ y = <.x, A>.)} e. (J Cn (J X.t J)))
32		ttcnlem.2	. 2 |- F = {<.x, y>. | (x e. X /\ y = <.x, A>.)}
33	31, 32	syl5eqel 1975	1 |- ((J e. Top /\ A e. X) -> F e. (J Cn (J X.t J)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {csn 3044  <.cop 3046  U.cuni 3177  {copab 3395   _I cid 3582   X. cxp 3984   |` cres 3988  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857   X.t ctx 8930   Cn ccn 9028

This theorem is referenced by:  trhom 14983

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-map 5383  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030

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