Theorem iunfo 9361

Description: Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)

Hypothesis

Ref	Expression
iunfo.1	|- T = U_ x e. A ( { x } X. B )

Assertion

Ref	Expression
iunfo	|- ( 2nd |` T ) : T -onto-> U_ x e. A B

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   T( x)

Proof of Theorem iunfo

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

Step	Hyp	Ref	Expression
1		fo2nd 7189	. . . 4 |- 2nd : _V -onto-> _V
2		fof 6115	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
3		ffn 6045	. . . 4 |- ( 2nd : _V --> _V -> 2nd Fn _V )
4	1, 2, 3	mp2b 10	. . 3 |- 2nd Fn _V
5		ssv 3625	. . 3 |- T C_ _V
6		fnssres 6004	. . 3 |- ( ( 2nd Fn _V /\ T C_ _V ) -> ( 2nd |` T ) Fn T )
7	4, 5, 6	mp2an 708	. 2 |- ( 2nd |` T ) Fn T
8		df-ima 5127	. . 3 |- ( 2nd " T ) = ran ( 2nd |` T )
9		iunfo.1	. . . . . . . . . . 11 |- T = U_ x e. A ( { x } X. B )
10	9	eleq2i 2693	. . . . . . . . . 10 |- ( z e. T <-> z e. U_ x e. A ( { x } X. B ) )
11		eliun 4524	. . . . . . . . . 10 |- ( z e. U_ x e. A ( { x } X. B ) <-> E. x e. A z e. ( { x } X. B ) )
12	10, 11	bitri 264	. . . . . . . . 9 |- ( z e. T <-> E. x e. A z e. ( { x } X. B ) )
13		xp2nd 7199	. . . . . . . . . . 11 |- ( z e. ( { x } X. B ) -> ( 2nd ` z ) e. B )
14		eleq1 2689	. . . . . . . . . . 11 |- ( ( 2nd ` z ) = y -> ( ( 2nd ` z ) e. B <-> y e. B ) )
15	13, 14	syl5ib 234	. . . . . . . . . 10 |- ( ( 2nd ` z ) = y -> ( z e. ( { x } X. B ) -> y e. B ) )
16	15	reximdv 3016	. . . . . . . . 9 |- ( ( 2nd ` z ) = y -> ( E. x e. A z e. ( { x } X. B ) -> E. x e. A y e. B ) )
17	12, 16	syl5bi 232	. . . . . . . 8 |- ( ( 2nd ` z ) = y -> ( z e. T -> E. x e. A y e. B ) )
18	17	impcom 446	. . . . . . 7 |- ( ( z e. T /\ ( 2nd ` z ) = y ) -> E. x e. A y e. B )
19	18	rexlimiva 3028	. . . . . 6 |- ( E. z e. T ( 2nd ` z ) = y -> E. x e. A y e. B )
20		nfiu1 4550	. . . . . . . . 9 |- F/_ x U_ x e. A ( { x } X. B )
21	9, 20	nfcxfr 2762	. . . . . . . 8 |- F/_ x T
22		nfv 1843	. . . . . . . 8 |- F/ x ( 2nd ` z ) = y
23	21, 22	nfrex 3007	. . . . . . 7 |- F/ x E. z e. T ( 2nd ` z ) = y
24		ssiun2 4563	. . . . . . . . . . . 12 |- ( x e. A -> ( { x } X. B ) C_ U_ x e. A ( { x } X. B ) )
25	24	adantr 481	. . . . . . . . . . 11 |- ( ( x e. A /\ y e. B ) -> ( { x } X. B ) C_ U_ x e. A ( { x } X. B ) )
26		simpr 477	. . . . . . . . . . . 12 |- ( ( x e. A /\ y e. B ) -> y e. B )
27		vsnid 4209	. . . . . . . . . . . . 13 |- x e. { x }
28		opelxp 5146	. . . . . . . . . . . . 13 |- ( <. x , y >. e. ( { x } X. B ) <-> ( x e. { x } /\ y e. B ) )
29	27, 28	mpbiran 953	. . . . . . . . . . . 12 |- ( <. x , y >. e. ( { x } X. B ) <-> y e. B )
30	26, 29	sylibr 224	. . . . . . . . . . 11 |- ( ( x e. A /\ y e. B ) -> <. x , y >. e. ( { x } X. B ) )
31	25, 30	sseldd 3604	. . . . . . . . . 10 |- ( ( x e. A /\ y e. B ) -> <. x , y >. e. U_ x e. A ( { x } X. B ) )
32	31, 9	syl6eleqr 2712	. . . . . . . . 9 |- ( ( x e. A /\ y e. B ) -> <. x , y >. e. T )
33		vex 3203	. . . . . . . . . 10 |- x e. _V
34		vex 3203	. . . . . . . . . 10 |- y e. _V
35	33, 34	op2nd 7177	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
36		fveq2 6191	. . . . . . . . . . 11 |- ( z = <. x , y >. -> ( 2nd ` z ) = ( 2nd ` <. x , y >. ) )
37	36	eqeq1d 2624	. . . . . . . . . 10 |- ( z = <. x , y >. -> ( ( 2nd ` z ) = y <-> ( 2nd ` <. x , y >. ) = y ) )
38	37	rspcev 3309	. . . . . . . . 9 |- ( ( <. x , y >. e. T /\ ( 2nd ` <. x , y >. ) = y ) -> E. z e. T ( 2nd ` z ) = y )
39	32, 35, 38	sylancl 694	. . . . . . . 8 |- ( ( x e. A /\ y e. B ) -> E. z e. T ( 2nd ` z ) = y )
40	39	ex 450	. . . . . . 7 |- ( x e. A -> ( y e. B -> E. z e. T ( 2nd ` z ) = y ) )
41	23, 40	rexlimi 3024	. . . . . 6 |- ( E. x e. A y e. B -> E. z e. T ( 2nd ` z ) = y )
42	19, 41	impbii 199	. . . . 5 |- ( E. z e. T ( 2nd ` z ) = y <-> E. x e. A y e. B )
43		fvelimab 6253	. . . . . 6 |- ( ( 2nd Fn _V /\ T C_ _V ) -> ( y e. ( 2nd " T ) <-> E. z e. T ( 2nd ` z ) = y ) )
44	4, 5, 43	mp2an 708	. . . . 5 |- ( y e. ( 2nd " T ) <-> E. z e. T ( 2nd ` z ) = y )
45		eliun 4524	. . . . 5 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
46	42, 44, 45	3bitr4i 292	. . . 4 |- ( y e. ( 2nd " T ) <-> y e. U_ x e. A B )
47	46	eqriv 2619	. . 3 |- ( 2nd " T ) = U_ x e. A B
48	8, 47	eqtr3i 2646	. 2 |- ran ( 2nd |` T ) = U_ x e. A B
49		df-fo 5894	. 2 |- ( ( 2nd |` T ) : T -onto-> U_ x e. A B <-> ( ( 2nd |` T ) Fn T /\ ran ( 2nd |` T ) = U_ x e. A B ) )
50	7, 48, 49	mpbir2an 955	1 |- ( 2nd |` T ) : T -onto-> U_ x e. A B

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-2nd 7169

This theorem is referenced by:  iundomg  9363