Theorem tz7.44-1 4229

Description: The value of F at (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49.

Hypotheses

Ref	Expression
tz7.44.1	|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
tz7.44.2	|- F Fn On
tz7.44.3	|- (x e. On -> (F` x) = (G` (F |` x)))
tz7.44.4	|- A e. V

Assertion

Ref	Expression
tz7.44-1	|- (F` (/)) = A

Distinct variable groups:   x,y,A   x,F   x,G   y,H

Proof of Theorem tz7.44-1

Step	Hyp	Ref	Expression
1		0elon 3026	. . 3 |- (/) e. On
2		fveq2 3835	. . . . 5 |- (x = (/) -> (F` x) = (F` (/)))
3		reseq2 3456	. . . . . 6 |- (x = (/) -> (F |` x) = (F |` (/)))
4	3	fveq2d 3839	. . . . 5 |- (x = (/) -> (G` (F |` x)) = (G` (F |` (/))))
5	2, 4	eqeq12d 1532	. . . 4 |- (x = (/) -> ((F` x) = (G` (F |` x)) <-> (F` (/)) = (G` (F |` (/)))))
6		tz7.44.3	. . . 4 |- (x e. On -> (F` x) = (G` (F |` x)))
7	5, 6	vtoclga 1898	. . 3 |- ((/) e. On -> (F` (/)) = (G` (F |` (/))))
8	1, 7	ax-mp 7	. 2 |- (F` (/)) = (G` (F |` (/)))
9		res0 3458	. . 3 |- (F |` (/)) = (/)
10	9	fveq2i 3838	. 2 |- (G` (F |` (/))) = (G` (/))
11		tz7.44.1	. . . 4 |- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
12	11	tz7.44lem1 4228	. . 3 |- Fun G
13		3mix1 821	. . . . . 6 |- ((x = (/) /\ y = A) -> ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x)))
14	13	ssopab2i 2901	. . . . 5 |- {<.x, y>. | (x = (/) /\ y = A)} (_ {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
15	14, 11	sseqtr4i 2146	. . . 4 |- {<.x, y>. | (x = (/) /\ y = A)} (_ G
16		0ex 2785	. . . . . 6 |- (/) e. V
17		tz7.44.4	. . . . . 6 |- A e. V
18		eqeq1 1524	. . . . . . 7 |- (x = (/) -> (x = (/) <-> (/) = (/)))
19	18	anbi1d 620	. . . . . 6 |- (x = (/) -> ((x = (/) /\ y = A) <-> ((/) = (/) /\ y = A)))
20		eqeq1 1524	. . . . . . 7 |- (y = A -> (y = A <-> A = A))
21	20	anbi2d 619	. . . . . 6 |- (y = A -> (((/) = (/) /\ y = A) <-> ((/) = (/) /\ A = A)))
22	16, 17, 19, 21	opelopab 2897	. . . . 5 |- (<.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)} <-> ((/) = (/) /\ A = A))
23		eqid 1518	. . . . 5 |- (/) = (/)
24		eqid 1518	. . . . 5 |- A = A
25	22, 23, 24	mpbir2an 735	. . . 4 |- <.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)}
26	15, 25	sselii 2118	. . 3 |- <.(/), A>. e. G
27	17	funopfv 3862	. . 3 |- (Fun G -> (<.(/), A>. e. G -> (G` (/)) = A))
28	12, 26, 27	mp2 43	. 2 |- (G` (/)) = A
29	8, 10, 28	3eqtri 1542	1 |- (F` (/)) = A

Syntax hints:  -. wn 2   -> wi 3   \/ wo 220   /\ wa 221   \/ w3o 780   = wceq 992   e. wcel 994  Vcvv 1857  (/)c0 2332  <.cop 2469  U.cuni 2569  {copab 2740  Oncon0 2975  Lim wlim 2976  dom cdm 3251  ran crn 3252   |` cres 3253  Fun wfun 3257   Fn wfn 3258  ` cfv 3263

This theorem is referenced by:  rdg0 4242

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fv 3279

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