Theorem tz7.44lem1 3922

Description: G is a function. Lemma for tz7.44-1 3923, tz7.44-2 3924, and tz7.44-3 3925.

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 3544	. . 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 3727	. . . 4 |- (H` (x` U.dom x)) e. V
3		visset 1810	. . . . 5 |- x e. V
4		rnexg 3355	. . . . . 6 |- (x e. V -> ran x e. V)
5		uniexg 2867	. . . . . 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 3023	. . . . . 6 |- -. Lim (/)
9		dm0 3319	. . . . . . 7 |- dom (/) = (/)
10		limeq 2956	. . . . . . 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 3307	. . . . . . 7 |- (x = (/) -> dom x = dom (/))
14		limeq 2956	. . . . . . 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 1918	. . 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 987	. 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 3531	. . 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 773   = wceq 955   e. wcel 957  E*wmo 1380  Vcvv 1808  (/)c0 2277  U.cuni 2499  {copab 2662  Lim wlim 2945  dom cdm 3166  ran crn 3167  Fun wfun 3172  ` cfv 3178

This theorem is referenced by:  tz7.44-1 3923  tz7.44-2 3924  tz7.44-3 3925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-lim 2949  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fv 3194

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