Theorem tz7.44lem1 3933

Description: G is a function. Lemma for tz7.44-1 3934, tz7.44-2 3935, and tz7.44-3 3936.

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 3554	. . 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 3738	. . . 4 |- (H` (x` U.dom x)) e. V
3		visset 1816	. . . . 5 |- x e. V
4		rnexg 3365	. . . . . 6 |- (x e. V -> ran x e. V)
5		uniexg 2877	. . . . . 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 3033	. . . . . 6 |- -. Lim (/)
9		dm0 3329	. . . . . . 7 |- dom (/) = (/)
10		limeq 2966	. . . . . . 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 3317	. . . . . . 7 |- (x = (/) -> dom x = dom (/))
14		limeq 2966	. . . . . . 7 |- (dom x = dom (/) -> (Lim dom x <-> Lim dom (/)))
15	13, 14	syl 10	. . . . . 6 |- (x = (/) -> (Lim dom x <-> Lim dom (/)))
16	15	biimpa 418	. . . . 5 |- ((x = (/) /\ Lim dom x) -> Lim dom (/))
17	12, 16	mto 106	. . . 4 |- -. (x = (/) /\ Lim dom x)
18	2, 7, 17	moeq3 1924	. . 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 990	. 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 3541	. . 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 776   = wceq 958   e. wcel 960  E*wmo 1383  Vcvv 1814  (/)c0 2283  U.cuni 2507  {copab 2671  Lim wlim 2955  dom cdm 3176  ran crn 3177  Fun wfun 3182  ` cfv 3188

This theorem is referenced by:  tz7.44-1 3934  tz7.44-2 3935  tz7.44-3 3936

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-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-lim 2959  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fv 3204

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