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Theorem tx2cn 23114
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 8000 . . 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 6718 . . . . . . . 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 7060 . . . . . . . 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 6911 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (2nd β€˜π‘§))
76eleq1d 2819 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (2nd β€˜π‘§) ∈ 𝑀))
8 1st2nd2 8014 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp1st 8007 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘§) ∈ 𝑋)
10 elxp6 8009 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑋 ∧ (2nd β€˜π‘§) ∈ 𝑀)))
11 anass 470 . . . . . . . . . . . 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 278 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑋) ∧ (2nd β€˜π‘§) ∈ 𝑀))
1312baib 537 . . . . . . . . . 10 ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑋) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (2nd β€˜π‘§) ∈ 𝑀))
148, 9, 13syl2anc 585 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) ↔ (2nd β€˜π‘§) ∈ 𝑀))
157, 14bitr4d 282 . . . . . . . 8 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((2nd β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ 𝑧 ∈ (𝑋 Γ— 𝑀)))
1615pm5.32i 576 . . . . . . 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 275 . . . . . 6 (𝑧 ∈ (β—‘(2nd β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑋 Γ— 𝑀)))
18 toponss 22429 . . . . . . . . . 10 ((𝑆 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑀 ∈ 𝑆) β†’ 𝑀 βŠ† π‘Œ)
1918adantll 713 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ 𝑀 βŠ† π‘Œ)
20 xpss2 5697 . . . . . . . . 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 3982 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑧 ∈ (𝑋 Γ— 𝑀) β†’ 𝑧 ∈ (𝑋 Γ— π‘Œ)))
2322pm4.71rd 564 . . . . . 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 290 . . . . 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 2731 . . . 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 22428 . . . . . 6 (𝑅 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝑅)
27 txopn 23106 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑀 ∈ 𝑆)) β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆))
2827expr 458 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑋 ∈ 𝑅) β†’ (𝑀 ∈ 𝑆 β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆)))
2926, 28mpidan 688 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑀 ∈ 𝑆 β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆)))
3029imp 408 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑆) β†’ (𝑋 Γ— 𝑀) ∈ (𝑅 Γ—t 𝑆))
3125, 30eqeltrd 2834 . . 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 3147 . 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 23095 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
34 iscn 22739 . . 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 589 . 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 712 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  βŸ¨cop 4635   Γ— cxp 5675  β—‘ccnv 5676   β†Ύ cres 5679   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  TopOnctopon 22412   Cn ccn 22728   Γ—t ctx 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cn 22731  df-tx 23066
This theorem is referenced by:  txcn  23130  txcmpb  23148  txkgen  23156  cnmpt2nd  23173  sxbrsiga  33289  txsconnlem  34231  txsconn  34232  hausgraph  41954
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