Theorem tz7.44lem1 7501

Description: G is a function. Lemma for tz7.44-1 7502, tz7.44-2 7503, and tz7.44-3 7504. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

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   y, A   y, H

Allowed substitution hints:   A( x)   G( x, y)   H( x)

Proof of Theorem tz7.44lem1

Step	Hyp	Ref	Expression
1		funopab 5923	. . 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. x E* y ( ( x = (/) /\ y = A ) \/ ( -. ( x = (/) \/ Lim dom x ) /\ y = ( H ` ( x ` U. dom x ) ) ) \/ ( Lim dom x /\ y = U. ran x ) ) )
2		fvex 6201	. . . 4 |- ( H ` ( x ` U. dom x ) ) e. _V
3		vex 3203	. . . . 5 |- x e. _V
4		rnexg 7098	. . . . 5 |- ( x e. _V -> ran x e. _V )
5		uniexg 6955	. . . . 5 |- ( ran x e. _V -> U. ran x e. _V )
6	3, 4, 5	mp2b 10	. . . 4 |- U. ran x e. _V
7		nlim0 5783	. . . . . 6 |- -. Lim (/)
8		dm0 5339	. . . . . . 7 |- dom (/) = (/)
9		limeq 5735	. . . . . . 7 |- ( dom (/) = (/) -> ( Lim dom (/) <-> Lim (/) ) )
10	8, 9	ax-mp 5	. . . . . 6 |- ( Lim dom (/) <-> Lim (/) )
11	7, 10	mtbir 313	. . . . 5 |- -. Lim dom (/)
12		dmeq 5324	. . . . . . 7 |- ( x = (/) -> dom x = dom (/) )
13		limeq 5735	. . . . . . 7 |- ( dom x = dom (/) -> ( Lim dom x <-> Lim dom (/) ) )
14	12, 13	syl 17	. . . . . 6 |- ( x = (/) -> ( Lim dom x <-> Lim dom (/) ) )
15	14	biimpa 501	. . . . 5 |- ( ( x = (/) /\ Lim dom x ) -> Lim dom (/) )
16	11, 15	mto 188	. . . 4 |- -. ( x = (/) /\ Lim dom x )
17	2, 6, 16	moeq3 3383	. . 3 |- E* y ( ( x = (/) /\ y = A ) \/ ( -. ( x = (/) \/ Lim dom x ) /\ y = ( H ` ( x ` U. dom x ) ) ) \/ ( Lim dom x /\ y = U. ran x ) )
18	1, 17	mpgbir 1726	. 2 |- Fun { <. x , y >. | ( ( x = (/) /\ y = A ) \/ ( -. ( x = (/) \/ Lim dom x ) /\ y = ( H ` ( x ` U. dom x ) ) ) \/ ( Lim dom x /\ y = U. ran x ) ) }
19		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 ) ) }
20	19	funeqi 5909	. 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 ) ) } )
21	18, 20	mpbir 221	1 |- Fun G

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   E*wmo 2471   _Vcvv 3200   (/)c0 3915   U.cuni 4436   {copab 4712   dom cdm 5114   ran crn 5115   Lim wlim 5724   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-lim 5728  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by: (None)