Theorem tx2ndc 21454

Description: The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
tx2ndc	|- ( ( R e. 2ndc /\ S e. 2ndc ) -> ( R tX S ) e. 2ndc )

Proof of Theorem tx2ndc

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

Step	Hyp	Ref	Expression
1		is2ndc 21249	. 2 |- ( R e. 2ndc <-> E. r e. TopBases ( r ~<_ om /\ ( topGen ` r ) = R ) )
2		is2ndc 21249	. 2 |- ( S e. 2ndc <-> E. s e. TopBases ( s ~<_ om /\ ( topGen ` s ) = S ) )
3		reeanv 3107	. . 3 |- ( E. r e. TopBases E. s e. TopBases ( ( r ~<_ om /\ ( topGen ` r ) = R ) /\ ( s ~<_ om /\ ( topGen ` s ) = S ) ) <-> ( E. r e. TopBases ( r ~<_ om /\ ( topGen ` r ) = R ) /\ E. s e. TopBases ( s ~<_ om /\ ( topGen ` s ) = S ) ) )
4		an4 865	. . . . 5 |- ( ( ( r ~<_ om /\ ( topGen ` r ) = R ) /\ ( s ~<_ om /\ ( topGen ` s ) = S ) ) <-> ( ( r ~<_ om /\ s ~<_ om ) /\ ( ( topGen ` r ) = R /\ ( topGen ` s ) = S ) ) )
5		txbasval 21409	. . . . . . . . . 10 |- ( ( r e. TopBases /\ s e. TopBases ) -> ( ( topGen ` r ) tX ( topGen ` s ) ) = ( r tX s ) )
6		eqid 2622	. . . . . . . . . . 11 |- ran ( x e. r , y e. s |-> ( x X. y ) ) = ran ( x e. r , y e. s |-> ( x X. y ) )
7	6	txval 21367	. . . . . . . . . 10 |- ( ( r e. TopBases /\ s e. TopBases ) -> ( r tX s ) = ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
8	5, 7	eqtrd 2656	. . . . . . . . 9 |- ( ( r e. TopBases /\ s e. TopBases ) -> ( ( topGen ` r ) tX ( topGen ` s ) ) = ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
9	8	adantr 481	. . . . . . . 8 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( ( topGen ` r ) tX ( topGen ` s ) ) = ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
10	6	txbas 21370	. . . . . . . . . 10 |- ( ( r e. TopBases /\ s e. TopBases ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) e. TopBases )
11	10	adantr 481	. . . . . . . . 9 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) e. TopBases )
12		omelon 8543	. . . . . . . . . . . 12 |- om e. On
13		vex 3203	. . . . . . . . . . . . . . . 16 |- s e. _V
14	13	xpdom1 8059	. . . . . . . . . . . . . . 15 |- ( r ~<_ om -> ( r X. s ) ~<_ ( om X. s ) )
15		omex 8540	. . . . . . . . . . . . . . . 16 |- om e. _V
16	15	xpdom2 8055	. . . . . . . . . . . . . . 15 |- ( s ~<_ om -> ( om X. s ) ~<_ ( om X. om ) )
17		domtr 8009	. . . . . . . . . . . . . . 15 |- ( ( ( r X. s ) ~<_ ( om X. s ) /\ ( om X. s ) ~<_ ( om X. om ) ) -> ( r X. s ) ~<_ ( om X. om ) )
18	14, 16, 17	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( r ~<_ om /\ s ~<_ om ) -> ( r X. s ) ~<_ ( om X. om ) )
19	18	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( r X. s ) ~<_ ( om X. om ) )
20		xpomen 8838	. . . . . . . . . . . . 13 |- ( om X. om ) ~~ om
21		domentr 8015	. . . . . . . . . . . . 13 |- ( ( ( r X. s ) ~<_ ( om X. om ) /\ ( om X. om ) ~~ om ) -> ( r X. s ) ~<_ om )
22	19, 20, 21	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( r X. s ) ~<_ om )
23		ondomen 8860	. . . . . . . . . . . 12 |- ( ( om e. On /\ ( r X. s ) ~<_ om ) -> ( r X. s ) e. dom card )
24	12, 22, 23	sylancr 695	. . . . . . . . . . 11 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( r X. s ) e. dom card )
25		eqid 2622	. . . . . . . . . . . . . 14 |- ( x e. r , y e. s |-> ( x X. y ) ) = ( x e. r , y e. s |-> ( x X. y ) )
26		vex 3203	. . . . . . . . . . . . . . 15 |- x e. _V
27		vex 3203	. . . . . . . . . . . . . . 15 |- y e. _V
28	26, 27	xpex 6962	. . . . . . . . . . . . . 14 |- ( x X. y ) e. _V
29	25, 28	fnmpt2i 7239	. . . . . . . . . . . . 13 |- ( x e. r , y e. s |-> ( x X. y ) ) Fn ( r X. s )
30	29	a1i 11	. . . . . . . . . . . 12 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( x e. r , y e. s |-> ( x X. y ) ) Fn ( r X. s ) )
31		dffn4 6121	. . . . . . . . . . . 12 |- ( ( x e. r , y e. s |-> ( x X. y ) ) Fn ( r X. s ) <-> ( x e. r , y e. s |-> ( x X. y ) ) : ( r X. s ) -onto-> ran ( x e. r , y e. s |-> ( x X. y ) ) )
32	30, 31	sylib 208	. . . . . . . . . . 11 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( x e. r , y e. s |-> ( x X. y ) ) : ( r X. s ) -onto-> ran ( x e. r , y e. s |-> ( x X. y ) ) )
33		fodomnum 8880	. . . . . . . . . . 11 |- ( ( r X. s ) e. dom card -> ( ( x e. r , y e. s |-> ( x X. y ) ) : ( r X. s ) -onto-> ran ( x e. r , y e. s |-> ( x X. y ) ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ ( r X. s ) ) )
34	24, 32, 33	sylc 65	. . . . . . . . . 10 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ ( r X. s ) )
35		domtr 8009	. . . . . . . . . 10 |- ( ( ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ ( r X. s ) /\ ( r X. s ) ~<_ om ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ om )
36	34, 22, 35	syl2anc 693	. . . . . . . . 9 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ om )
37		2ndci 21251	. . . . . . . . 9 |- ( ( ran ( x e. r , y e. s |-> ( x X. y ) ) e. TopBases /\ ran ( x e. r , y e. s |-> ( x X. y ) ) ~<_ om ) -> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) e. 2ndc )
38	11, 36, 37	syl2anc 693	. . . . . . . 8 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) e. 2ndc )
39	9, 38	eqeltrd 2701	. . . . . . 7 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( ( topGen ` r ) tX ( topGen ` s ) ) e. 2ndc )
40		oveq12 6659	. . . . . . . 8 |- ( ( ( topGen ` r ) = R /\ ( topGen ` s ) = S ) -> ( ( topGen ` r ) tX ( topGen ` s ) ) = ( R tX S ) )
41	40	eleq1d 2686	. . . . . . 7 |- ( ( ( topGen ` r ) = R /\ ( topGen ` s ) = S ) -> ( ( ( topGen ` r ) tX ( topGen ` s ) ) e. 2ndc <-> ( R tX S ) e. 2ndc ) )
42	39, 41	syl5ibcom 235	. . . . . 6 |- ( ( ( r e. TopBases /\ s e. TopBases ) /\ ( r ~<_ om /\ s ~<_ om ) ) -> ( ( ( topGen ` r ) = R /\ ( topGen ` s ) = S ) -> ( R tX S ) e. 2ndc ) )
43	42	expimpd 629	. . . . 5 |- ( ( r e. TopBases /\ s e. TopBases ) -> ( ( ( r ~<_ om /\ s ~<_ om ) /\ ( ( topGen ` r ) = R /\ ( topGen ` s ) = S ) ) -> ( R tX S ) e. 2ndc ) )
44	4, 43	syl5bi 232	. . . 4 |- ( ( r e. TopBases /\ s e. TopBases ) -> ( ( ( r ~<_ om /\ ( topGen ` r ) = R ) /\ ( s ~<_ om /\ ( topGen ` s ) = S ) ) -> ( R tX S ) e. 2ndc ) )
45	44	rexlimivv 3036	. . 3 |- ( E. r e. TopBases E. s e. TopBases ( ( r ~<_ om /\ ( topGen ` r ) = R ) /\ ( s ~<_ om /\ ( topGen ` s ) = S ) ) -> ( R tX S ) e. 2ndc )
46	3, 45	sylbir 225	. 2 |- ( ( E. r e. TopBases ( r ~<_ om /\ ( topGen ` r ) = R ) /\ E. s e. TopBases ( s ~<_ om /\ ( topGen ` s ) = S ) ) -> ( R tX S ) e. 2ndc )
47	1, 2, 46	syl2anb 496	1 |- ( ( R e. 2ndc /\ S e. 2ndc ) -> ( R tX S ) e. 2ndc )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   Oncon0 5723   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761   topGenctg 16098   TopBasesctb 20749   2ndcc2ndc 21241   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-topgen 16104  df-bases 20750  df-2ndc 21243  df-tx 21365

This theorem is referenced by: (None)