Theorem tz7.44lem1 3912

Description: G is a function. Lemma for tz7.44-1 3913, tz7.44-2 3914, and tz7.44-3 3915.

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 3534	. . 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 3717	. . . 4 |- (H` (x` U.dom x)) e. V
3		visset 1804	. . . . 5 |- x e. V
4		rnexg 3345	. . . . . 6 |- (x e. V -> ran x e. V)
5		uniexg 2862	. . . . . 6 |- (ran x e. V -> U.ran x e. V)
6	4, 5	syl 10	. . . . 5 |- (x e. V -> U.ran x e. V)
7	3, 6	ax-mp 7	. . . 4 |- U.ran x e. V
8		nlim0 3017	. . . . . 6 |- -. Lim (/)
9		dm0 3312	. . . . . . 7 |- dom (/) = (/)
10		limeq 2950	. . . . . . 7 |- (dom (/) = (/) -> (Lim dom (/) <-> Lim (/)))
11	9, 10	ax-mp 7	. . . . . 6 |- (Lim dom (/) <-> Lim (/))
12	8, 11	mtbir 192	. . . . 5 |- -. Lim dom (/)
13		dmeq 3300	. . . . . . 7 |- (x = (/) -> dom x = dom (/))
14		limeq 2950	. . . . . . 7 |- (dom x = dom (/) -> (Lim dom x <-> Lim dom (/)))
15	13, 14	syl 10	. . . . . 6 |- (x = (/) -> (Lim dom x <-> Lim dom (/)))
16	15	biimpa 416	. . . . 5 |- ((x = (/) /\ Lim dom x) -> Lim dom (/))
17	12, 16	mto 106	. . . 4 |- -. (x = (/) /\ Lim dom x)
18	2, 7, 17	moeq3 1912	. . 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 985	. 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 3521	. . 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 190	1 |- Fun G

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  E*wmo 1374  Vcvv 1802  (/)c0 2270  U.cuni 2493  {copab 2656  Lim wlim 2939  dom cdm 3160  ran crn 3161  Fun wfun 3166  ` cfv 3172

This theorem is referenced by:  tz7.44-1 3913  tz7.44-2 3914  tz7.44-3 3915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-lim 2943  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fv 3188

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