Theorem th3qcor 6410

Description: Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

Ref	Expression
th3q.1	|- .~ e. _V
th3q.2	|- .~ Er ( S X. S )
th3q.4	|- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
th3q.5	|- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }

Assertion

Ref	Expression
th3qcor	|- Fun G

Distinct variable groups:   x, y, z, w, v, u, t, s, f, g, h, .~   x, S, y, z, w, v, u, t, s, f, g, h   x, .+ , y, z, w, v, u, t, s, f, g, h

Allowed substitution hints:   G( x, y, z, w, v, u, t, f, g, h, s)

Proof of Theorem th3qcor

Step	Hyp	Ref	Expression
1		th3q.1	. . . . 5 |- .~ e. _V
2		th3q.2	. . . . 5 |- .~ Er ( S X. S )
3		th3q.4	. . . . 5 |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
4	1, 2, 3	th3qlem2 6409	. . . 4 |- ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
5		moanimv 2024	. . . 4 |- ( E* z ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) <-> ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
6	4, 5	mpbir 145	. . 3 |- E* z ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
7	6	funoprab 5759	. 2 |- Fun { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }
8		th3q.5	. . 3 |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }
9	8	funeqi 5049	. 2 |- ( Fun G <-> Fun { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) } )
10	7, 9	mpbir 145	1 |- Fun G

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   E.wex 1427   e. wcel 1439   E*wmo 1950   _Vcvv 2620   <.cop 3453   class class class wbr 3851   X. cxp 4449   Fun wfun 5022  (class class class)co 5666   {coprab 5667   Er wer 6303   [cec 6304   /.cqs 6305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-er 6306  df-ec 6308  df-qs 6312

This theorem is referenced by: (None)