Theorem tz7.44lem1 5135

Description: G is a function. Lemma for tz7.44-1 5136, tz7.44-2 5137, and tz7.44-3 5138.

Hypothesis

Ref	Expression
tz7.44lem1.1	|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}

Assertion

Ref	Expression
tz7.44lem1	|- Fun G

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

Proof of Theorem tz7.44lem1

Step	Hyp	Ref	Expression
1		funopab 4455	. . 3 |- (Fun {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))} <-> A.xE*y((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x)))
2		fvex 4689	. . . 4 |- (H` (x` U.dom x)) e. _V
3		visset 2295	. . . . 5 |- x e. _V
4		rnexg 4207	. . . . . 6 |- (x e. _V -> ran x e. _V)
5		uniexg 3795	. . . . . 6 |- (ran x e. _V -> U.ran x e. _V)
6	4, 5	syl 12	. . . . 5 |- (x e. _V -> U.ran x e. _V)
7	3, 6	ax-mp 7	. . . 4 |- U.ran x e. _V
8		nlim0 3721	. . . . . 6 |- -. Lim (/)
9		dm0 4170	. . . . . . 7 |- dom (/) = (/)
10		limeq 3669	. . . . . . 7 |- (dom (/) = (/) -> (Lim dom (/) <-> Lim (/)))
11	9, 10	ax-mp 7	. . . . . 6 |- (Lim dom (/) <-> Lim (/))
12	8, 11	mtbir 209	. . . . 5 |- -. Lim dom (/)
13		dmeq 4157	. . . . . . 7 |- (x = (/) -> dom x = dom (/))
14		limeq 3669	. . . . . . 7 |- (dom x = dom (/) -> (Lim dom x <-> Lim dom (/)))
15	13, 14	syl 12	. . . . . 6 |- (x = (/) -> (Lim dom x <-> Lim dom (/)))
16	15	biimpa 460	. . . . 5 |- ((x = (/) /\ Lim dom x) -> Lim dom (/))
17	12, 16	mto 121	. . . 4 |- -. (x = (/) /\ Lim dom x)
18	2, 7, 17	moeq3 2432	. . 3 |- E*y((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))
19	1, 18	mpgbir 1334	. 2 |- Fun {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
20		tz7.44lem1.1	. . 3 |- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
21		funeq 4441	. . 3 |- (G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))} -> (Fun G <-> Fun {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}))
22	20, 21	ax-mp 7	. 2 |- (Fun G <-> Fun {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))})
23	19, 22	mpbir 207	1 |- Fun G

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300  E*wmo 1772  _Vcvv 2292  (/)c0 2875  U.cuni 3177  {copab 3395  Lim wlim 3658  dom cdm 3986  ran crn 3987  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  tz7.44-1 5136  tz7.44-2 5137  tz7.44-3 5138

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-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-3or 859  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-lim 3662  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fv 4014

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