Theorem txval 21367

Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Hypothesis

Ref	Expression
txval.1	|- B = ran ( x e. R , y e. S |-> ( x X. y ) )

Assertion

Ref	Expression
txval	|- ( ( R e. V /\ S e. W ) -> ( R tX S ) = ( topGen ` B ) )

Distinct variable groups:   x, y, R   x, S, y

Allowed substitution hints:   B( x, y)   V( x, y)   W( x, y)

Proof of Theorem txval

Dummy variables r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( R e. V -> R e. _V )
2		elex 3212	. 2 |- ( S e. W -> S e. _V )
3		mpt2eq12 6715	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. r , y e. s |-> ( x X. y ) ) = ( x e. R , y e. S |-> ( x X. y ) ) )
4	3	rneqd 5353	. . . . 5 |- ( ( r = R /\ s = S ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) = ran ( x e. R , y e. S |-> ( x X. y ) ) )
5		txval.1	. . . . 5 |- B = ran ( x e. R , y e. S |-> ( x X. y ) )
6	4, 5	syl6eqr 2674	. . . 4 |- ( ( r = R /\ s = S ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) = B )
7	6	fveq2d 6195	. . 3 |- ( ( r = R /\ s = S ) -> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) = ( topGen ` B ) )
8		df-tx 21365	. . 3 |- tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
9		fvex 6201	. . 3 |- ( topGen ` B ) e. _V
10	7, 8, 9	ovmpt2a 6791	. 2 |- ( ( R e. _V /\ S e. _V ) -> ( R tX S ) = ( topGen ` B ) )
11	1, 2, 10	syl2an 494	1 |- ( ( R e. V /\ S e. W ) -> ( R tX S ) = ( topGen ` B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   topGenctg 16098   tX ctx 21363

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tx 21365

This theorem is referenced by:  eltx  21371  txtop  21372  txtopon  21394  txopn  21405  txss12  21408  txbasval  21409  txcnp  21423  txcnmpt  21427  txrest  21434  txlm  21451  tx2ndc  21454  txflf  21810  mbfimaopnlem  23422