Theorem tz7.44-1 2966

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 group(s):   x,y,A   x,F   x,G   y,H

Proof of Theorem tz7.44-1

Step	Hyp	Ref	Expression
1		0elon 2277	. . 3 |- (/) e. On
2		fveq2 2832	. . . . 5 |- (x = (/) -> (F` x) = (F` (/)))
3		reseq2 2576	. . . . . 6 |- (x = (/) -> (F |` x) = (F |` (/)))
4	3	fveq2d 2836	. . . . 5 |- (x = (/) -> (G` (F |` x)) = (G` (F |` (/))))
5	2, 4	cleq12d 1115	. . . 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 1387	. . 3 |- ((/) e. On -> (F` (/)) = (G` (F |` (/))))
8	1, 7	ax-mp 6	. 2 |- (F` (/)) = (G` (F |` (/)))
9		res0 2578	. . 3 |- (F |` (/)) = (/)
10	9	fveq2i 2835	. 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 2965	. . 3 |- Fun G
13		3mix1 600	. . . . . 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 2120	. . . . 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	sseqtr4 1533	. . . 4 |- {<.x, y>. | (x = (/) /\ y = A)} (_ G
16		cleqid 1102	. . . . . 6 |- (/) = (/)
17		cleqid 1102	. . . . . 6 |- A = A
18	16, 17	pm3.2i 234	. . . . 5 |- ((/) = (/) /\ A = A)
19		0ex 1745	. . . . . 6 |- (/) e. V
20		tz7.44.4	. . . . . 6 |- A e. V
21		cleq1 1107	. . . . . . 7 |- (x = (/) -> (x = (/) <-> (/) = (/)))
22	21	anbi1d 469	. . . . . 6 |- (x = (/) -> ((x = (/) /\ y = A) <-> ((/) = (/) /\ y = A)))
23		cleq1 1107	. . . . . . 7 |- (y = A -> (y = A <-> A = A))
24	23	anbi2d 468	. . . . . 6 |- (y = A -> (((/) = (/) /\ y = A) <-> ((/) = (/) /\ A = A)))
25	19, 20, 22, 24	opelopab 2117	. . . . 5 |- (<.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)} <-> ((/) = (/) /\ A = A))
26	18, 25	mpbir 165	. . . 4 |- <.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)}
27	15, 26	sselii 1505	. . 3 |- <.(/), A>. e. G
28	20	funfvopi 2853	. . 3 |- (Fun G -> (<.(/), A>. e. G -> (G` (/)) = A))
29	12, 27, 28	mp2 43	. 2 |- (G` (/)) = A
30	8, 10, 29	3eqtr 1123	1 |- (F` (/)) = A

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  <.cop 1810  U.cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   |` cres 2412  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  rdgzer 2979

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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