Theorem omp1eomlem 7384

Description: Lemma for omp1eom 7385. (Contributed by Jim Kingdon, 11-Jul-2023.)

Hypotheses

Ref	Expression
omp1eom.f	|- F = ( x e. om |-> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )
omp1eom.s	|- S = ( x e. om |-> suc x )
omp1eom.g	|- G = case ( S , ( _I |` 1o ) )

Assertion

Ref	Expression
omp1eomlem	|- F : om -1-1-onto-> ( om ⊔ 1o )

Distinct variable group:   x, G

Allowed substitution hints:   S( x)   F( x)

Proof of Theorem omp1eomlem

Dummy variables z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omp1eom.f	. . 3 |- F = ( x e. om |-> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )
2		el1o 6669	. . . . . . 7 |- ( x e. 1o <-> x = (/) )
3	2	biimpri 133	. . . . . 6 |- ( x = (/) -> x e. 1o )
4	3	adantl 277	. . . . 5 |- ( ( ( T. /\ x e. om ) /\ x = (/) ) -> x e. 1o )
5		djurcl 7342	. . . . 5 |- ( x e. 1o -> (inr ` x ) e. ( om ⊔ 1o ) )
6	4, 5	syl 14	. . . 4 |- ( ( ( T. /\ x e. om ) /\ x = (/) ) -> (inr ` x ) e. ( om ⊔ 1o ) )
7		nnpredcl 4744	. . . . . 6 |- ( x e. om -> U. x e. om )
8	7	ad2antlr 489	. . . . 5 |- ( ( ( T. /\ x e. om ) /\ -. x = (/) ) -> U. x e. om )
9		djulcl 7341	. . . . 5 |- ( U. x e. om -> (inl ` U. x ) e. ( om ⊔ 1o ) )
10	8, 9	syl 14	. . . 4 |- ( ( ( T. /\ x e. om ) /\ -. x = (/) ) -> (inl ` U. x ) e. ( om ⊔ 1o ) )
11		nndceq0 4739	. . . . 5 |- ( x e. om -> DECID x = (/) )
12	11	adantl 277	. . . 4 |- ( ( T. /\ x e. om ) -> DECID x = (/) )
13	6, 10, 12	ifcldadc 3651	. . 3 |- ( ( T. /\ x e. om ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) e. ( om ⊔ 1o ) )
14		omp1eom.s	. . . . . . . 8 |- S = ( x e. om |-> suc x )
15		peano2 4716	. . . . . . . 8 |- ( x e. om -> suc x e. om )
16	14, 15	fmpti 5828	. . . . . . 7 |- S : om --> om
17	16	a1i 9	. . . . . 6 |- ( T. -> S : om --> om )
18		f1oi 5653	. . . . . . . . 9 |- ( _I |` 1o ) : 1o -1-1-onto-> 1o
19		f1of 5613	. . . . . . . . 9 |- ( ( _I |` 1o ) : 1o -1-1-onto-> 1o -> ( _I |` 1o ) : 1o --> 1o )
20	18, 19	ax-mp 5	. . . . . . . 8 |- ( _I |` 1o ) : 1o --> 1o
21		1onn 6752	. . . . . . . . 9 |- 1o e. om
22		omelon 4730	. . . . . . . . . 10 |- om e. On
23	22	onelssi 4549	. . . . . . . . 9 |- ( 1o e. om -> 1o C_ om )
24	21, 23	ax-mp 5	. . . . . . . 8 |- 1o C_ om
25		fss 5520	. . . . . . . 8 |- ( ( ( _I |` 1o ) : 1o --> 1o /\ 1o C_ om ) -> ( _I |` 1o ) : 1o --> om )
26	20, 24, 25	mp2an 426	. . . . . . 7 |- ( _I |` 1o ) : 1o --> om
27	26	a1i 9	. . . . . 6 |- ( T. -> ( _I |` 1o ) : 1o --> om )
28	17, 27	casef 7378	. . . . 5 |- ( T. -> case ( S , ( _I |` 1o ) ) : ( om ⊔ 1o ) --> om )
29		omp1eom.g	. . . . . 6 |- G = case ( S , ( _I |` 1o ) )
30	29	feq1i 5500	. . . . 5 |- ( G : ( om ⊔ 1o ) --> om <-> case ( S , ( _I |` 1o ) ) : ( om ⊔ 1o ) --> om )
31	28, 30	sylibr 134	. . . 4 |- ( T. -> G : ( om ⊔ 1o ) --> om )
32	31	ffvelcdmda 5811	. . 3 |- ( ( T. /\ y e. ( om ⊔ 1o ) ) -> ( G ` y ) e. om )
33		ffn 5507	. . . . . . . . . . . . . . . 16 |- ( S : om --> om -> S Fn om )
34	16, 33	mp1i 10	. . . . . . . . . . . . . . 15 |- ( ( z e. om /\ y = (inl ` z ) ) -> S Fn om )
35		ffun 5510	. . . . . . . . . . . . . . . 16 |- ( ( _I |` 1o ) : 1o --> 1o -> Fun ( _I |` 1o ) )
36	20, 35	mp1i 10	. . . . . . . . . . . . . . 15 |- ( ( z e. om /\ y = (inl ` z ) ) -> Fun ( _I |` 1o ) )
37		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( z e. om /\ y = (inl ` z ) ) -> z e. om )
38	34, 36, 37	caseinl 7381	. . . . . . . . . . . . . 14 |- ( ( z e. om /\ y = (inl ` z ) ) -> (case ( S , ( _I |` 1o ) ) ` (inl ` z ) ) = ( S ` z ) )
39	29	eqcomi 2236	. . . . . . . . . . . . . . . 16 |- case ( S , ( _I |` 1o ) ) = G
40	39	a1i 9	. . . . . . . . . . . . . . 15 |- ( ( z e. om /\ y = (inl ` z ) ) -> case ( S , ( _I |` 1o ) ) = G )
41		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( z e. om /\ y = (inl ` z ) ) -> y = (inl ` z ) )
42	41	eqcomd 2238	. . . . . . . . . . . . . . 15 |- ( ( z e. om /\ y = (inl ` z ) ) -> (inl ` z ) = y )
43	40, 42	fveq12d 5676	. . . . . . . . . . . . . 14 |- ( ( z e. om /\ y = (inl ` z ) ) -> (case ( S , ( _I |` 1o ) ) ` (inl ` z ) ) = ( G ` y ) )
44		peano2 4716	. . . . . . . . . . . . . . . 16 |- ( z e. om -> suc z e. om )
45		suceq 4522	. . . . . . . . . . . . . . . . 17 |- ( x = z -> suc x = suc z )
46	45, 14	fvmptg 5752	. . . . . . . . . . . . . . . 16 |- ( ( z e. om /\ suc z e. om ) -> ( S ` z ) = suc z )
47	44, 46	mpdan 421	. . . . . . . . . . . . . . 15 |- ( z e. om -> ( S ` z ) = suc z )
48	47	adantr 276	. . . . . . . . . . . . . 14 |- ( ( z e. om /\ y = (inl ` z ) ) -> ( S ` z ) = suc z )
49	38, 43, 48	3eqtr3d 2273	. . . . . . . . . . . . 13 |- ( ( z e. om /\ y = (inl ` z ) ) -> ( G ` y ) = suc z )
50		peano3 4717	. . . . . . . . . . . . . 14 |- ( z e. om -> suc z =/= (/) )
51	50	adantr 276	. . . . . . . . . . . . 13 |- ( ( z e. om /\ y = (inl ` z ) ) -> suc z =/= (/) )
52	49, 51	eqnetrd 2436	. . . . . . . . . . . 12 |- ( ( z e. om /\ y = (inl ` z ) ) -> ( G ` y ) =/= (/) )
53	52	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( G ` y ) =/= (/) )
54	53	necomd 2498	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> (/) =/= ( G ` y ) )
55	54	neneqd 2433	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> -. (/) = ( G ` y ) )
56		simplr 529	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> x = (/) )
57	56	eqeq1d 2241	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( x = ( G ` y ) <-> (/) = ( G ` y ) ) )
58	55, 57	mtbird 680	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> -. x = ( G ` y ) )
59		djune 7368	. . . . . . . . . . . 12 |- ( ( z e. _V /\ x e. _V ) -> (inl ` z ) =/= (inr ` x ) )
60	59	elvd 2817	. . . . . . . . . . 11 |- ( z e. _V -> (inl ` z ) =/= (inr ` x ) )
61	60	elv 2816	. . . . . . . . . 10 |- (inl ` z ) =/= (inr ` x )
62	61	neii 2414	. . . . . . . . 9 |- -. (inl ` z ) = (inr ` x )
63		simprr 533	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> y = (inl ` z ) )
64		simpr 110	. . . . . . . . . . . 12 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> x = (/) )
65	64	iftrued 3628	. . . . . . . . . . 11 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) = (inr ` x ) )
66	65	adantr 276	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) = (inr ` x ) )
67	63, 66	eqeq12d 2247	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) <-> (inl ` z ) = (inr ` x ) ) )
68	62, 67	mtbiri 682	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> -. y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )
69	58, 68	2falsed 710	. . . . . . 7 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
70	69	rexlimdvaa 2661	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> ( E. z e. om y = (inl ` z ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) ) )
71		simplr 529	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> x = (/) )
72	29	a1i 9	. . . . . . . . . . . 12 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> G = case ( S , ( _I |` 1o ) ) )
73		simpr 110	. . . . . . . . . . . 12 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> y = (inr ` z ) )
74	72, 73	fveq12d 5676	. . . . . . . . . . 11 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> ( G ` y ) = (case ( S , ( _I |` 1o ) ) ` (inr ` z ) ) )
75	14	funmpt2 5390	. . . . . . . . . . . . . 14 |- Fun S
76	75	a1i 9	. . . . . . . . . . . . 13 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> Fun S )
77		fnresi 5475	. . . . . . . . . . . . . 14 |- ( _I |` 1o ) Fn 1o
78	77	a1i 9	. . . . . . . . . . . . 13 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> ( _I |` 1o ) Fn 1o )
79		simpl 109	. . . . . . . . . . . . 13 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> z e. 1o )
80	76, 78, 79	caseinr 7382	. . . . . . . . . . . 12 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> (case ( S , ( _I |` 1o ) ) ` (inr ` z ) ) = ( ( _I |` 1o ) ` z ) )
81		fvresi 5876	. . . . . . . . . . . . 13 |- ( z e. 1o -> ( ( _I |` 1o ) ` z ) = z )
82	81	adantr 276	. . . . . . . . . . . 12 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> ( ( _I |` 1o ) ` z ) = z )
83	80, 82	eqtrd 2265	. . . . . . . . . . 11 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> (case ( S , ( _I |` 1o ) ) ` (inr ` z ) ) = z )
84		el1o 6669	. . . . . . . . . . . 12 |- ( z e. 1o <-> z = (/) )
85	79, 84	sylib 122	. . . . . . . . . . 11 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> z = (/) )
86	74, 83, 85	3eqtrd 2269	. . . . . . . . . 10 |- ( ( z e. 1o /\ y = (inr ` z ) ) -> ( G ` y ) = (/) )
87	86	adantl 277	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( G ` y ) = (/) )
88	71, 87	eqtr4d 2268	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> x = ( G ` y ) )
89	85	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> z = (/) )
90	71, 89	eqtr4d 2268	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> x = z )
91	90	fveq2d 5673	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> (inr ` x ) = (inr ` z ) )
92	65	adantr 276	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) = (inr ` x ) )
93		simprr 533	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> y = (inr ` z ) )
94	91, 92, 93	3eqtr4rd 2276	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )
95	88, 94	2thd 175	. . . . . . 7 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
96	95	rexlimdvaa 2661	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> ( E. z e. 1o y = (inr ` z ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) ) )
97		djur 7359	. . . . . . . 8 |- ( y e. ( om ⊔ 1o ) <-> ( E. z e. om y = (inl ` z ) \/ E. z e. 1o y = (inr ` z ) ) )
98	97	biimpi 120	. . . . . . 7 |- ( y e. ( om ⊔ 1o ) -> ( E. z e. om y = (inl ` z ) \/ E. z e. 1o y = (inr ` z ) ) )
99	98	ad2antlr 489	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> ( E. z e. om y = (inl ` z ) \/ E. z e. 1o y = (inr ` z ) ) )
100	70, 96, 99	mpjaod 726	. . . . 5 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ x = (/) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
101		simplll 535	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> x e. om )
102		simplr 529	. . . . . . . . . . . 12 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> -. x = (/) )
103	102	neqned 2419	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> x =/= (/) )
104		nnsucpred 4738	. . . . . . . . . . 11 |- ( ( x e. om /\ x =/= (/) ) -> suc U. x = x )
105	101, 103, 104	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> suc U. x = x )
106	105	eqeq2d 2244	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( suc z = suc U. x <-> suc z = x ) )
107		eqcom 2234	. . . . . . . . 9 |- ( suc z = x <-> x = suc z )
108	106, 107	bitrdi 196	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( suc z = suc U. x <-> x = suc z ) )
109		simprr 533	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> y = (inl ` z ) )
110		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> -. x = (/) )
111	110	iffalsed 3631	. . . . . . . . . . . 12 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) = (inl ` U. x ) )
112	111	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) = (inl ` U. x ) )
113	109, 112	eqeq12d 2247	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) <-> (inl ` z ) = (inl ` U. x ) ) )
114		vuniex 4558	. . . . . . . . . . . 12 |- U. x e. _V
115		inl11 7355	. . . . . . . . . . . 12 |- ( ( z e. _V /\ U. x e. _V ) -> ( (inl ` z ) = (inl ` U. x ) <-> z = U. x ) )
116	114, 115	mpan2 425	. . . . . . . . . . 11 |- ( z e. _V -> ( (inl ` z ) = (inl ` U. x ) <-> z = U. x ) )
117	116	elv 2816	. . . . . . . . . 10 |- ( (inl ` z ) = (inl ` U. x ) <-> z = U. x )
118	113, 117	bitrdi 196	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) <-> z = U. x ) )
119		nnon 4731	. . . . . . . . . . 11 |- ( z e. om -> z e. On )
120	119	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> z e. On )
121	7	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> U. x e. om )
122		nnon 4731	. . . . . . . . . . 11 |- ( U. x e. om -> U. x e. On )
123	121, 122	syl 14	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> U. x e. On )
124		suc11 4679	. . . . . . . . . 10 |- ( ( z e. On /\ U. x e. On ) -> ( suc z = suc U. x <-> z = U. x ) )
125	120, 123, 124	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( suc z = suc U. x <-> z = U. x ) )
126	118, 125	bitr4d 191	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) <-> suc z = suc U. x ) )
127	49	adantl 277	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( G ` y ) = suc z )
128	127	eqeq2d 2244	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( x = ( G ` y ) <-> x = suc z ) )
129	108, 126, 128	3bitr4rd 221	. . . . . . 7 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. om /\ y = (inl ` z ) ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
130	129	rexlimdvaa 2661	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> ( E. z e. om y = (inl ` z ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) ) )
131		simplr 529	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> -. x = (/) )
132	86	adantl 277	. . . . . . . . . 10 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( G ` y ) = (/) )
133	132	eqeq2d 2244	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( x = ( G ` y ) <-> x = (/) ) )
134	131, 133	mtbird 680	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> -. x = ( G ` y ) )
135		djune 7368	. . . . . . . . . . . 12 |- ( ( U. x e. _V /\ z e. _V ) -> (inl ` U. x ) =/= (inr ` z ) )
136	135	elvd 2817	. . . . . . . . . . 11 |- ( U. x e. _V -> (inl ` U. x ) =/= (inr ` z ) )
137	114, 136	ax-mp 5	. . . . . . . . . 10 |- (inl ` U. x ) =/= (inr ` z )
138	137	nesymi 2458	. . . . . . . . 9 |- -. (inr ` z ) = (inl ` U. x )
139	73, 111	eqeqan12rd 2249	. . . . . . . . 9 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) <-> (inr ` z ) = (inl ` U. x ) ) )
140	138, 139	mtbiri 682	. . . . . . . 8 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> -. y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )
141	134, 140	2falsed 710	. . . . . . 7 |- ( ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) /\ ( z e. 1o /\ y = (inr ` z ) ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
142	141	rexlimdvaa 2661	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> ( E. z e. 1o y = (inr ` z ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) ) )
143	98	ad2antlr 489	. . . . . 6 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> ( E. z e. om y = (inl ` z ) \/ E. z e. 1o y = (inr ` z ) ) )
144	130, 142, 143	mpjaod 726	. . . . 5 |- ( ( ( x e. om /\ y e. ( om ⊔ 1o ) ) /\ -. x = (/) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
145		exmiddc 844	. . . . . . 7 |- (DECID x = (/) -> ( x = (/) \/ -. x = (/) ) )
146	11, 145	syl 14	. . . . . 6 |- ( x e. om -> ( x = (/) \/ -. x = (/) ) )
147	146	adantr 276	. . . . 5 |- ( ( x e. om /\ y e. ( om ⊔ 1o ) ) -> ( x = (/) \/ -. x = (/) ) )
148	100, 144, 147	mpjaodan 806	. . . 4 |- ( ( x e. om /\ y e. ( om ⊔ 1o ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
149	148	adantl 277	. . 3 |- ( ( T. /\ ( x e. om /\ y e. ( om ⊔ 1o ) ) ) -> ( x = ( G ` y ) <-> y = if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) ) )
150	1, 13, 32, 149	f1o2d 6259	. 2 |- ( T. -> F : om -1-1-onto-> ( om ⊔ 1o ) )
151	150	mptru 1407	1 |- F : om -1-1-onto-> ( om ⊔ 1o )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   T. wtru 1399   e. wcel 2203   =/= wne 2412   E.wrex 2521   _Vcvv 2812   C_ wss 3210   (/)c0 3507   ifcif 3619   U.cuni 3913   |-> cmpt 4170   _I cid 4408   Oncon0 4483   suc csuc 4485   omcom 4711   |` cres 4750   Fun wfun 5345   Fn wfn 5346   -->wf 5347   -1-1-onto->wf1o 5350   ` cfv 5351   1oc1o 6639   ⊔ cdju 7327  inlcinl 7335  inrcinr 7336  casecdjucase 7373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1st 6333  df-2nd 6334  df-1o 6646  df-dju 7328  df-inl 7337  df-inr 7338  df-case 7374

This theorem is referenced by:  omp1eom  7385