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Theorem tx2cn 23113
Description: Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx2cn ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
Proof of Theorem tx2cn
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f2ndres 7999 . . 3 (2nd β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘Œ
21a1i 11 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘Œ)
3 ffn 6717 . . . . . . . 8 ((2nd β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘Œ β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ))
4 elpreima 7059 . . . . . . . 8 ((2nd β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀)))
51, 3, 4mp2b 10 . . . . . . 7 (𝑧 ∈ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀))
6 fvres 6910 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (2nd β€˜π‘§))
76eleq1d 2818 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (2nd β€˜π‘§) ∈ 𝑀))
8 1st2nd2 8013 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp1st 8006 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘§) ∈ 𝑋)
10 elxp6 8008 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑋 ∧ (2nd β€˜π‘§) ∈ 𝑀)))
11 anass 469 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑋) ∧ (2nd β€˜π‘§) ∈ 𝑀) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑋 ∧ (2nd β€˜π‘§) ∈ 𝑀)))
1210, 11bitr4i 277 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑋) ∧ (2nd β€˜π‘§) ∈ 𝑀))
1312baib 536 . . . . . . . . . 10 ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑋) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (2nd β€˜π‘§) ∈ 𝑀))
148, 9, 13syl2anc 584 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (2nd β€˜π‘§) ∈ 𝑀))
157, 14bitr4d 281 . . . . . . . 8 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ 𝑧 ∈ (𝑋 Γ— 𝑀)))
1615pm5.32i 575 . . . . . . 7 ((𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑋 Γ— 𝑀)))
175, 16bitri 274 . . . . . 6 (𝑧 ∈ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑋 Γ— 𝑀)))
18 toponss 22428 . . . . . . . . . 10 ((𝑆 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑀 ∈ 𝑆) β†’ 𝑀 βŠ† π‘Œ)
1918adantll 712 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ 𝑀 βŠ† π‘Œ)
20 xpss2 5696 . . . . . . . . 9 (𝑀 βŠ† π‘Œ β†’ (𝑋 Γ— 𝑀) βŠ† (𝑋 Γ— π‘Œ))
2119, 20syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑋 Γ— 𝑀) βŠ† (𝑋 Γ— π‘Œ))
2221sseld 3981 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) β†’ 𝑧 ∈ (𝑋 Γ— π‘Œ)))
2322pm4.71rd 563 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑋 Γ— 𝑀))))
2417, 23bitr4id 289 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑧 ∈ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ 𝑧 ∈ (𝑋 Γ— 𝑀)))
2524eqrdv 2730 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) = (𝑋 Γ— 𝑀))
26 toponmax 22427 . . . . . 6 (𝑅 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝑅)
27 txopn 23105 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑀 ∈ 𝑆)) β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆))
2827expr 457 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑋 ∈ 𝑅) β†’ (𝑀 ∈ 𝑆 β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆)))
2926, 28mpidan 687 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑀 ∈ 𝑆 β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆)))
3029imp 407 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆))
3125, 30eqeltrd 2833 . . 3 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))
3231ralrimiva 3146 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ βˆ€π‘€ ∈ 𝑆 (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))
33 txtopon 23094 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
34 iscn 22738 . . 3 (((𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ ((2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆) ↔ ((2nd β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘Œ ∧ βˆ€π‘€ ∈ 𝑆 (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))))
3533, 34sylancom 588 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ ((2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆) ↔ ((2nd β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘Œ ∧ βˆ€π‘€ ∈ 𝑆 (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))))
362, 32, 35mpbir2and 711 1 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  βŸ¨cop 4634   Γ— cxp 5674  β—‘ccnv 5675   β†Ύ cres 5678   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  TopOnctopon 22411   Cn ccn 22727   Γ—t ctx 23063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cn 22730  df-tx 23065
This theorem is referenced by:  txcn  23129  txcmpb  23147  txkgen  23155  cnmpt2nd  23172  sxbrsiga  33284  txsconnlem  34226  txsconn  34227  hausgraph  41944
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