Theorem ctiunct 12491

Description: A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each B ( x ): it refers to B ( x ) together with the G ( x ) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be ( ph /\ x e. A ) -> E. g g : om -onto-> ( B ⊔ 1o ). This is almost omiunct 12495 (which uses countable choice) although that is for a countably infinite collection not any countable collection.

Compare with the case of two sets instead of countably many, as seen at unct 12493, which says that the union of two countable sets is countable .

The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12446) and using the first number to map to an element x of A and the second number to map to an element of B(x) . In this way we are able to map to every element of U_ x e. A B. Although it would be possible to work directly with countability expressed as F : om -onto-> ( A ⊔ 1o ), we instead use functions from subsets of the natural numbers via ctssdccl 7140 and ctssdc 7142.

(Contributed by Jim Kingdon, 31-Oct-2023.)

Hypotheses

Ref	Expression
ctiunct.a	|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )
ctiunct.b	|- ( ( ph /\ x e. A ) -> G : om -onto-> ( B ⊔ 1o ) )

Assertion

Ref	Expression
ctiunct	|- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )

Distinct variable groups:   A, h, x   B, h   x, F   ph, x

Allowed substitution hints:   ph( h)   B( x)   F( h)   G( x, h)

Proof of Theorem ctiunct

Dummy variables j k n u z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpomen 12446	. . . . 5 |- ( om X. om ) ~~ om
2	1	ensymi 6808	. . . 4 |- om ~~ ( om X. om )
3		bren 6773	. . . 4 |- ( om ~~ ( om X. om ) <-> E. j j : om -1-1-onto-> ( om X. om ) )
4	2, 3	mpbi 145	. . 3 |- E. j j : om -1-1-onto-> ( om X. om )
5	4	a1i 9	. 2 |- ( ph -> E. j j : om -1-1-onto-> ( om X. om ) )
6		ctiunct.a	. . . . . . . 8 |- ( ph -> F : om -onto-> ( A ⊔ 1o ) )
7		eqid 2189	. . . . . . . 8 |- { w e. om | ( F ` w ) e. (inl " A ) } = { w e. om | ( F ` w ) e. (inl " A ) }
8		eqid 2189	. . . . . . . 8 |- ( `'inl o. F ) = ( `'inl o. F )
9	6, 7, 8	ctssdccl 7140	. . . . . . 7 |- ( ph -> ( { w e. om | ( F ` w ) e. (inl " A ) } C_ om /\ ( `'inl o. F ) : { w e. om | ( F ` w ) e. (inl " A ) } -onto-> A /\ A. n e. om DECID n e. { w e. om | ( F ` w ) e. (inl " A ) } ) )
10	9	simp1d 1011	. . . . . 6 |- ( ph -> { w e. om | ( F ` w ) e. (inl " A ) } C_ om )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> { w e. om | ( F ` w ) e. (inl " A ) } C_ om )
12	9	simp3d 1013	. . . . . 6 |- ( ph -> A. n e. om DECID n e. { w e. om | ( F ` w ) e. (inl " A ) } )
13	12	adantr 276	. . . . 5 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> A. n e. om DECID n e. { w e. om | ( F ` w ) e. (inl " A ) } )
14	9	simp2d 1012	. . . . . 6 |- ( ph -> ( `'inl o. F ) : { w e. om | ( F ` w ) e. (inl " A ) } -onto-> A )
15	14	adantr 276	. . . . 5 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> ( `'inl o. F ) : { w e. om | ( F ` w ) e. (inl " A ) } -onto-> A )
16		ctiunct.b	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> G : om -onto-> ( B ⊔ 1o ) )
17		eqid 2189	. . . . . . . 8 |- { w e. om | ( G ` w ) e. (inl " B ) } = { w e. om | ( G ` w ) e. (inl " B ) }
18		eqid 2189	. . . . . . . 8 |- ( `'inl o. G ) = ( `'inl o. G )
19	16, 17, 18	ctssdccl 7140	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( { w e. om | ( G ` w ) e. (inl " B ) } C_ om /\ ( `'inl o. G ) : { w e. om | ( G ` w ) e. (inl " B ) } -onto-> B /\ A. n e. om DECID n e. { w e. om | ( G ` w ) e. (inl " B ) } ) )
20	19	simp1d 1011	. . . . . 6 |- ( ( ph /\ x e. A ) -> { w e. om | ( G ` w ) e. (inl " B ) } C_ om )
21	20	adantlr 477	. . . . 5 |- ( ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) /\ x e. A ) -> { w e. om | ( G ` w ) e. (inl " B ) } C_ om )
22	19	simp3d 1013	. . . . . 6 |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. { w e. om | ( G ` w ) e. (inl " B ) } )
23	22	adantlr 477	. . . . 5 |- ( ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) /\ x e. A ) -> A. n e. om DECID n e. { w e. om | ( G ` w ) e. (inl " B ) } )
24	19	simp2d 1012	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( `'inl o. G ) : { w e. om | ( G ` w ) e. (inl " B ) } -onto-> B )
25	24	adantlr 477	. . . . 5 |- ( ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) /\ x e. A ) -> ( `'inl o. G ) : { w e. om | ( G ` w ) e. (inl " B ) } -onto-> B )
26		simpr 110	. . . . 5 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> j : om -1-1-onto-> ( om X. om ) )
27		eqid 2189	. . . . 5 |- { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) }
28	11, 13, 15, 21, 23, 25, 26, 27	ctiunctlemuom 12487	. . . 4 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } C_ om )
29		eqid 2189	. . . . . 6 |- ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) = ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) )
30		nfv 1539	. . . . . . . . 9 |- F/ x ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) }
31		nfcsb1v 3105	. . . . . . . . . 10 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) }
32	31	nfel2 2345	. . . . . . . . 9 |- F/ x ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) }
33	30, 32	nfan 1576	. . . . . . . 8 |- F/ x ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } )
34		nfcv 2332	. . . . . . . 8 |- F/_ x om
35	33, 34	nfrabxy 2671	. . . . . . 7 |- F/_ x { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) }
36		nfcsb1v 3105	. . . . . . . 8 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G )
37		nfcv 2332	. . . . . . . 8 |- F/_ x ( 2nd ` ( j ` n ) )
38	36, 37	nffv 5544	. . . . . . 7 |- F/_ x ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) )
39	35, 38	nfmpt 4110	. . . . . 6 |- F/_ x ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) )
40	11, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35	ctiunctlemfo 12490	. . . . 5 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B )
41		omex 4610	. . . . . . . 8 |- om e. _V
42	41	rabex 4162	. . . . . . 7 |- { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } e. _V
43	42	mptex 5763	. . . . . 6 |- ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) e. _V
44		foeq1 5453	. . . . . 6 |- ( k = ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) -> ( k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B <-> ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B ) )
45	43, 44	spcev 2847	. . . . 5 |- ( ( n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } |-> ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) ) ) : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B -> E. k k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B )
46	40, 45	syl 14	. . . 4 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> E. k k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B )
47	11, 13, 15, 21, 23, 25, 26, 27	ctiunctlemudc 12488	. . . 4 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> A. n e. om DECID n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } )
48		sseq1 3193	. . . . . 6 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( u C_ om <-> { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } C_ om ) )
49		foeq2 5454	. . . . . . 7 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( k : u -onto-> U_ x e. A B <-> k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B ) )
50	49	exbidv 1836	. . . . . 6 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( E. k k : u -onto-> U_ x e. A B <-> E. k k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B ) )
51		eleq2 2253	. . . . . . . 8 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( n e. u <-> n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } ) )
52	51	dcbid 839	. . . . . . 7 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> (DECID n e. u <-> DECID n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } ) )
53	52	ralbidv 2490	. . . . . 6 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( A. n e. om DECID n e. u <-> A. n e. om DECID n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } ) )
54	48, 50, 53	3anbi123d 1323	. . . . 5 |- ( u = { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -> ( ( u C_ om /\ E. k k : u -onto-> U_ x e. A B /\ A. n e. om DECID n e. u ) <-> ( { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } C_ om /\ E. k k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B /\ A. n e. om DECID n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } ) ) )
55	42, 54	spcev 2847	. . . 4 |- ( ( { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } C_ om /\ E. k k : { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } -onto-> U_ x e. A B /\ A. n e. om DECID n e. { z e. om | ( ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) } /\ ( 2nd ` ( j ` z ) ) e. [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) } ) } ) -> E. u ( u C_ om /\ E. k k : u -onto-> U_ x e. A B /\ A. n e. om DECID n e. u ) )
56	28, 46, 47, 55	syl3anc 1249	. . 3 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> E. u ( u C_ om /\ E. k k : u -onto-> U_ x e. A B /\ A. n e. om DECID n e. u ) )
57		ctssdc 7142	. . . 4 |- ( E. u ( u C_ om /\ E. k k : u -onto-> U_ x e. A B /\ A. n e. om DECID n e. u ) <-> E. k k : om -onto-> ( U_ x e. A B ⊔ 1o ) )
58		foeq1 5453	. . . . 5 |- ( k = h -> ( k : om -onto-> ( U_ x e. A B ⊔ 1o ) <-> h : om -onto-> ( U_ x e. A B ⊔ 1o ) ) )
59	58	cbvexv 1930	. . . 4 |- ( E. k k : om -onto-> ( U_ x e. A B ⊔ 1o ) <-> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
60	57, 59	bitri 184	. . 3 |- ( E. u ( u C_ om /\ E. k k : u -onto-> U_ x e. A B /\ A. n e. om DECID n e. u ) <-> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
61	56, 60	sylib 122	. 2 |- ( ( ph /\ j : om -1-1-onto-> ( om X. om ) ) -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
62	5, 61	exlimddv 1910	1 |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 835   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   {crab 2472   [_csb 3072   C_ wss 3144   U_ciun 3901   class class class wbr 4018   |-> cmpt 4079   omcom 4607   X. cxp 4642   `'ccnv 4643   "cima 4647   o. ccom 4648   -onto->wfo 5233   -1-1-onto->wf1o 5234   ` cfv 5235   1stc1st 6163   2ndc2nd 6164   1oc1o 6434   ~~ cen 6764   ⊔ cdju 7066  inlcinl 7074

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-1o 6441  df-er 6559  df-en 6767  df-dju 7067  df-inl 7076  df-inr 7077  df-case 7113  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-dvds 11827

This theorem is referenced by:  ctiunctal  12492  unct  12493