Theorem tz7.44-1 3934

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 3028	. . 3 |- (/) e. On
2		fveq2 3730	. . . . 5 |- (x = (/) -> (F` x) = (F` (/)))
3		reseq2 3375	. . . . . 6 |- (x = (/) -> (F |` x) = (F |` (/)))
4	3	fveq2d 3734	. . . . 5 |- (x = (/) -> (G` (F |` x)) = (G` (F |` (/))))
5	2, 4	eqeq12d 1492	. . . 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 1855	. . 3 |- ((/) e. On -> (F` (/)) = (G` (F |` (/))))
8	1, 7	ax-mp 7	. 2 |- (F` (/)) = (G` (F |` (/)))
9		res0 3377	. . 3 |- (F |` (/)) = (/)
10	9	fveq2i 3733	. 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 3933	. . 3 |- Fun G
13		3mix1 817	. . . . . 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 2829	. . . . 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 2097	. . . 4 |- {<.x, y>. | (x = (/) /\ y = A)} (_ G
16		0ex 2716	. . . . . 6 |- (/) e. V
17		tz7.44.4	. . . . . 6 |- A e. V
18		eqeq1 1484	. . . . . . 7 |- (x = (/) -> (x = (/) <-> (/) = (/)))
19	18	anbi1d 619	. . . . . 6 |- (x = (/) -> ((x = (/) /\ y = A) <-> ((/) = (/) /\ y = A)))
20		eqeq1 1484	. . . . . . 7 |- (y = A -> (y = A <-> A = A))
21	20	anbi2d 618	. . . . . 6 |- (y = A -> (((/) = (/) /\ y = A) <-> ((/) = (/) /\ A = A)))
22	16, 17, 19, 21	opelopab 2826	. . . . 5 |- (<.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)} <-> ((/) = (/) /\ A = A))
23		eqid 1478	. . . . 5 |- (/) = (/)
24		eqid 1478	. . . . 5 |- A = A
25	22, 23, 24	mpbir2an 732	. . . 4 |- <.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)}
26	15, 25	sselii 2069	. . 3 |- <.(/), A>. e. G
27	17	funopfv 3757	. . 3 |- (Fun G -> (<.(/), A>. e. G -> (G` (/)) = A))
28	12, 26, 27	mp2 43	. 2 |- (G` (/)) = A
29	8, 10, 28	3eqtr 1502	1 |- (F` (/)) = A

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960  Vcvv 1814  (/)c0 2283  <.cop 2415  U.cuni 2507  {copab 2671  Oncon0 2954  Lim wlim 2955  dom cdm 3176  ran crn 3177   |` cres 3178  Fun wfun 3182   Fn wfn 3183  ` cfv 3188

This theorem is referenced by:  rdg0 3947

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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