Theorem ovmptss 7258

Description: If all the values of the mapping are subsets of a class X, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
ovmptss.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
ovmptss	|- ( A. x e. A A. y e. B C C_ X -> ( E F G ) C_ X )

Distinct variable groups:   x, y, A   y, B   x, X, y

Allowed substitution hints:   B( x)   C( x, y)   E( x, y)   F( x, y)   G( x, y)

Proof of Theorem ovmptss

Dummy variables v u z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovmptss.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
2		mpt2mptsx 7233	. . . 4 |- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
3	1, 2	eqtri 2644	. . 3 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
4	3	fvmptss 6292	. 2 |- ( A. z e. U_ x e. A ( { x } X. B ) [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X -> ( F ` <. E , G >. ) C_ X )
5		vex 3203	. . . . . . . 8 |- u e. _V
6		vex 3203	. . . . . . . 8 |- v e. _V
7	5, 6	op1std 7178	. . . . . . 7 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
8	7	csbeq1d 3540	. . . . . 6 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
9	5, 6	op2ndd 7179	. . . . . . . 8 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
10	9	csbeq1d 3540	. . . . . . 7 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
11	10	csbeq2dv 3992	. . . . . 6 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	8, 11	eqtrd 2656	. . . . 5 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
13	12	sseq1d 3632	. . . 4 |- ( z = <. u , v >. -> ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X <-> [_ u / x ]_ [_ v / y ]_ C C_ X ) )
14	13	raliunxp 5261	. . 3 |- ( A. z e. U_ u e. A ( { u } X. [_ u / x ]_ B ) [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X <-> A. u e. A A. v e. [_ u / x ]_ B [_ u / x ]_ [_ v / y ]_ C C_ X )
15		nfcv 2764	. . . . 5 |- F/_ u ( { x } X. B )
16		nfcv 2764	. . . . . 6 |- F/_ x { u }
17		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ u / x ]_ B
18	16, 17	nfxp 5142	. . . . 5 |- F/_ x ( { u } X. [_ u / x ]_ B )
19		sneq 4187	. . . . . 6 |- ( x = u -> { x } = { u } )
20		csbeq1a 3542	. . . . . 6 |- ( x = u -> B = [_ u / x ]_ B )
21	19, 20	xpeq12d 5140	. . . . 5 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
22	15, 18, 21	cbviun 4557	. . . 4 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
23	22	raleqi 3142	. . 3 |- ( A. z e. U_ x e. A ( { x } X. B ) [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X <-> A. z e. U_ u e. A ( { u } X. [_ u / x ]_ B ) [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X )
24		nfv 1843	. . . 4 |- F/ u A. y e. B C C_ X
25		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
26		nfcv 2764	. . . . . 6 |- F/_ x X
27	25, 26	nfss 3596	. . . . 5 |- F/ x [_ u / x ]_ [_ v / y ]_ C C_ X
28	17, 27	nfral 2945	. . . 4 |- F/ x A. v e. [_ u / x ]_ B [_ u / x ]_ [_ v / y ]_ C C_ X
29		nfv 1843	. . . . . 6 |- F/ v C C_ X
30		nfcsb1v 3549	. . . . . . 7 |- F/_ y [_ v / y ]_ C
31		nfcv 2764	. . . . . . 7 |- F/_ y X
32	30, 31	nfss 3596	. . . . . 6 |- F/ y [_ v / y ]_ C C_ X
33		csbeq1a 3542	. . . . . . 7 |- ( y = v -> C = [_ v / y ]_ C )
34	33	sseq1d 3632	. . . . . 6 |- ( y = v -> ( C C_ X <-> [_ v / y ]_ C C_ X ) )
35	29, 32, 34	cbvral 3167	. . . . 5 |- ( A. y e. B C C_ X <-> A. v e. B [_ v / y ]_ C C_ X )
36		csbeq1a 3542	. . . . . . 7 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
37	36	sseq1d 3632	. . . . . 6 |- ( x = u -> ( [_ v / y ]_ C C_ X <-> [_ u / x ]_ [_ v / y ]_ C C_ X ) )
38	20, 37	raleqbidv 3152	. . . . 5 |- ( x = u -> ( A. v e. B [_ v / y ]_ C C_ X <-> A. v e. [_ u / x ]_ B [_ u / x ]_ [_ v / y ]_ C C_ X ) )
39	35, 38	syl5bb 272	. . . 4 |- ( x = u -> ( A. y e. B C C_ X <-> A. v e. [_ u / x ]_ B [_ u / x ]_ [_ v / y ]_ C C_ X ) )
40	24, 28, 39	cbvral 3167	. . 3 |- ( A. x e. A A. y e. B C C_ X <-> A. u e. A A. v e. [_ u / x ]_ B [_ u / x ]_ [_ v / y ]_ C C_ X )
41	14, 23, 40	3bitr4ri 293	. 2 |- ( A. x e. A A. y e. B C C_ X <-> A. z e. U_ x e. A ( { x } X. B ) [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C C_ X )
42		df-ov 6653	. . 3 |- ( E F G ) = ( F ` <. E , G >. )
43	42	sseq1i 3629	. 2 |- ( ( E F G ) C_ X <-> ( F ` <. E , G >. ) C_ X )
44	4, 41, 43	3imtr4i 281	1 |- ( A. x e. A A. y e. B C C_ X -> ( E F G ) C_ X )

Syntax hints:   -> wi 4   = wceq 1483   A.wral 2912   [_csb 3533   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   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-csb 3534  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  relmpt2opab  7259  relxpchom  16821  reldv  23634