Theorem tz7.44lem1 2965

Description: G is a function. Lemma for tz7.44-1 2966, tz7.44-2 2967, and tz7.44-3 2968.

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 group(s):   x,y,A   x,G   y,H

Proof of Theorem tz7.44lem1

Step	Hyp	Ref	Expression
1		funopab 2694	. . 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 2838	. . . 4 |- (H` (x` U.dom x)) e. V
3		visset 1350	. . . . 5 |- x e. V
4		rnexg 2569	. . . . . 6 |- (x e. V -> ran x e. V)
5		uniexg 1948	. . . . . 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 6	. . . 4 |- U.ran x e. V
8		nlim0 2282	. . . . . 6 |- -. Lim (/)
9		dm0 2542	. . . . . . 7 |- dom (/) = (/)
10		limeq 2211	. . . . . . 7 |- (dom (/) = (/) -> (Lim dom (/) <-> Lim (/)))
11	9, 10	ax-mp 6	. . . . . 6 |- (Lim dom (/) <-> Lim (/))
12	8, 11	mtbir 167	. . . . 5 |- -. Lim dom (/)
13		dmeq 2531	. . . . . . 7 |- (x = (/) -> dom x = dom (/))
14		limeq 2211	. . . . . . 7 |- (dom x = dom (/) -> (Lim dom x <-> Lim dom (/)))
15	13, 14	syl 12	. . . . . 6 |- (x = (/) -> (Lim dom x <-> Lim dom (/)))
16	15	biimpa 324	. . . . 5 |- ((x = (/) /\ Lim dom x) -> Lim dom (/))
17	12, 16	mto 93	. . . 4 |- -. (x = (/) /\ Lim dom x)
18	2, 7, 17	moeq3 1432	. . 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 686	. 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 2683	. . 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 6	. 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 165	1 |- Fun G

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   /\ wa 196   \/ w3o 580  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  U.cuni 1919  {copab 2055  Lim wlim 2200  dom cdm 2410  ran crn 2411  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  tz7.44-1 2966  tz7.44-2 2967  tz7.44-3 2968

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-lim 2204  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fv 2438

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