Theorem ctiunct 13191

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 13195 (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 13193, 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 13146) 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 7402 and ctssdc 7404.

(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 13146	. . . . 5 |- ( om X. om ) ~~ om
2	1	ensymi 7022	. . . 4 |- om ~~ ( om X. om )
3		bren 6983	. . . 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 2232	. . . . . . . 8 |- { w e. om | ( F ` w ) e. (inl " A ) } = { w e. om | ( F ` w ) e. (inl " A ) }
8		eqid 2232	. . . . . . . 8 |- ( `'inl o. F ) = ( `'inl o. F )
9	6, 7, 8	ctssdccl 7402	. . . . . . 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 1036	. . . . . 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 1038	. . . . . 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 1037	. . . . . 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 2232	. . . . . . . 8 |- { w e. om | ( G ` w ) e. (inl " B ) } = { w e. om | ( G ` w ) e. (inl " B ) }
18		eqid 2232	. . . . . . . 8 |- ( `'inl o. G ) = ( `'inl o. G )
19	16, 17, 18	ctssdccl 7402	. . . . . . 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 1036	. . . . . 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 1038	. . . . . 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 1037	. . . . . 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 2232	. . . . 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 13187	. . . 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 2232	. . . . . 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 1577	. . . . . . . . 9 |- F/ x ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) }
31		nfcsb1v 3171	. . . . . . . . . 10 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) }
32	31	nfel2 2397	. . . . . . . . 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 1614	. . . . . . . 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 2384	. . . . . . . 8 |- F/_ x om
35	33, 34	nfrabw 2725	. . . . . . 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 3171	. . . . . . . 8 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G )
37		nfcv 2384	. . . . . . . 8 |- F/_ x ( 2nd ` ( j ` n ) )
38	36, 37	nffv 5680	. . . . . . 7 |- F/_ x ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) )
39	35, 38	nfmpt 4202	. . . . . 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 13190	. . . . 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 4715	. . . . . . . 8 |- om e. _V
42	41	rabex 4256	. . . . . . 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 5912	. . . . . 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 5586	. . . . . 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 2912	. . . . 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 13188	. . . 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 3261	. . . . . 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 5587	. . . . . . 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 1874	. . . . . 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 2296	. . . . . . . 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 846	. . . . . . 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 2542	. . . . . 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 1349	. . . . 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 2912	. . . 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 1274	. . 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 7404	. . . 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 5586	. . . . 5 |- ( k = h -> ( k : om -onto-> ( U_ x e. A B ⊔ 1o ) <-> h : om -onto-> ( U_ x e. A B ⊔ 1o ) ) )
59	58	cbvexv 1968	. . . 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 1948	1 |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   {crab 2524   [_csb 3138   C_ wss 3211   U_ciun 3991   class class class wbr 4109   |-> cmpt 4171   omcom 4712   X. cxp 4747   `'ccnv 4748   "cima 4752   o. ccom 4753   -onto->wfo 5350   -1-1-onto->wf1o 5351   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   1oc1o 6640   ~~ cen 6973   ⊔ cdju 7328  inlcinl 7336

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-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  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-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dju 7329  df-inl 7338  df-inr 7339  df-case 7375  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-dvds 12474

This theorem is referenced by:  ctiunctal  13192  unct  13193