Theorem ctiunct 12782

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 12786 (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 12784, 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 12737) 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 7212 and ctssdc 7214.

(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 12737	. . . . 5 |- ( om X. om ) ~~ om
2	1	ensymi 6873	. . . 4 |- om ~~ ( om X. om )
3		bren 6834	. . . 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 2204	. . . . . . . 8 |- { w e. om | ( F ` w ) e. (inl " A ) } = { w e. om | ( F ` w ) e. (inl " A ) }
8		eqid 2204	. . . . . . . 8 |- ( `'inl o. F ) = ( `'inl o. F )
9	6, 7, 8	ctssdccl 7212	. . . . . . 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 2204	. . . . . . . 8 |- { w e. om | ( G ` w ) e. (inl " B ) } = { w e. om | ( G ` w ) e. (inl " B ) }
18		eqid 2204	. . . . . . . 8 |- ( `'inl o. G ) = ( `'inl o. G )
19	16, 17, 18	ctssdccl 7212	. . . . . . 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 2204	. . . . 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 12778	. . . 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 2204	. . . . . 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 1550	. . . . . . . . 9 |- F/ x ( 1st ` ( j ` z ) ) e. { w e. om | ( F ` w ) e. (inl " A ) }
31		nfcsb1v 3125	. . . . . . . . . 10 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) }
32	31	nfel2 2360	. . . . . . . . 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 1587	. . . . . . . 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 2347	. . . . . . . 8 |- F/_ x om
35	33, 34	nfrabw 2686	. . . . . . 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 3125	. . . . . . . 8 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G )
37		nfcv 2347	. . . . . . . 8 |- F/_ x ( 2nd ` ( j ` n ) )
38	36, 37	nffv 5585	. . . . . . 7 |- F/_ x ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) )
39	35, 38	nfmpt 4135	. . . . . 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 12781	. . . . 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 4640	. . . . . . . 8 |- om e. _V
42	41	rabex 4187	. . . . . . 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 5809	. . . . . 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 5493	. . . . . 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 2867	. . . . 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 12779	. . . 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 3215	. . . . . 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 5494	. . . . . . 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 1847	. . . . . 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 2268	. . . . . . . 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 2505	. . . . . 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 1324	. . . . 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 2867	. . . 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 7214	. . . 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 5493	. . . . 5 |- ( k = h -> ( k : om -onto-> ( U_ x e. A B ⊔ 1o ) <-> h : om -onto-> ( U_ x e. A B ⊔ 1o ) ) )
59	58	cbvexv 1941	. . . 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 1921	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 1372   E.wex 1514   e. wcel 2175   A.wral 2483   {crab 2487   [_csb 3092   C_ wss 3165   U_ciun 3926   class class class wbr 4043   |-> cmpt 4104   omcom 4637   X. cxp 4672   `'ccnv 4673   "cima 4677   o. ccom 4678   -onto->wfo 5268   -1-1-onto->wf1o 5269   ` cfv 5270   1stc1st 6223   2ndc2nd 6224   1oc1o 6494   ~~ cen 6824   ⊔ cdju 7138  inlcinl 7146

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-xor 1395  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-er 6619  df-en 6827  df-dju 7139  df-inl 7148  df-inr 7149  df-case 7185  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-dvds 12070

This theorem is referenced by:  ctiunctal  12783  unct  12784