Theorem ctiunct 12600

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 12604 (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 12602, 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 12555) 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 7172 and ctssdc 7174.

(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 12555	. . . . 5 |- ( om X. om ) ~~ om
2	1	ensymi 6838	. . . 4 |- om ~~ ( om X. om )
3		bren 6803	. . . 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 2193	. . . . . . . 8 |- { w e. om | ( F ` w ) e. (inl " A ) } = { w e. om | ( F ` w ) e. (inl " A ) }
8		eqid 2193	. . . . . . . 8 |- ( `'inl o. F ) = ( `'inl o. F )
9	6, 7, 8	ctssdccl 7172	. . . . . . 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 2193	. . . . . . . 8 |- { w e. om | ( G ` w ) e. (inl " B ) } = { w e. om | ( G ` w ) e. (inl " B ) }
18		eqid 2193	. . . . . . . 8 |- ( `'inl o. G ) = ( `'inl o. G )
19	16, 17, 18	ctssdccl 7172	. . . . . . 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 2193	. . . . 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 12596	. . . 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 2193	. . . . . 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 3114	. . . . . . . . . 10 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` z ) ) ) / x ]_ { w e. om | ( G ` w ) e. (inl " B ) }
32	31	nfel2 2349	. . . . . . . . 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 2336	. . . . . . . 8 |- F/_ x om
35	33, 34	nfrabw 2675	. . . . . . 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 3114	. . . . . . . 8 |- F/_ x [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G )
37		nfcv 2336	. . . . . . . 8 |- F/_ x ( 2nd ` ( j ` n ) )
38	36, 37	nffv 5565	. . . . . . 7 |- F/_ x ( [_ ( ( `'inl o. F ) ` ( 1st ` ( j ` n ) ) ) / x ]_ ( `'inl o. G ) ` ( 2nd ` ( j ` n ) ) )
39	35, 38	nfmpt 4122	. . . . . 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 12599	. . . . 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 4626	. . . . . . . 8 |- om e. _V
42	41	rabex 4174	. . . . . . 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 5785	. . . . . 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 5473	. . . . . 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 2856	. . . . 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 12597	. . . 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 3203	. . . . . 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 5474	. . . . . . 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 2257	. . . . . . . 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 2494	. . . . . 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 2856	. . . 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 7174	. . . 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 5473	. . . . 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 2164   A.wral 2472   {crab 2476   [_csb 3081   C_ wss 3154   U_ciun 3913   class class class wbr 4030   |-> cmpt 4091   omcom 4623   X. cxp 4658   `'ccnv 4659   "cima 4663   o. ccom 4664   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   1oc1o 6464   ~~ cen 6794   ⊔ cdju 7098  inlcinl 7106

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-er 6589  df-en 6797  df-dju 7099  df-inl 7108  df-inr 7109  df-case 7145  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-dvds 11934

This theorem is referenced by:  ctiunctal  12601  unct  12602