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Theorem tx1cn 23104
Description: Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx1cn ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
Proof of Theorem tx1cn
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 7995 . . 3 (1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹
21a1i 11 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹)
3 ffn 6714 . . . . . . . 8 ((1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹ β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ))
4 elpreima 7056 . . . . . . . 8 ((1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀)))
51, 3, 4mp2b 10 . . . . . . 7 (𝑧 ∈ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀))
6 fvres 6907 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
76eleq1d 2818 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (1st β€˜π‘§) ∈ 𝑀))
8 1st2nd2 8010 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp2nd 8004 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘§) ∈ π‘Œ)
10 elxp6 8005 . . . . . . . . . . . 12 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑀 ∧ (2nd β€˜π‘§) ∈ π‘Œ)))
11 anass 469 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑀) ∧ (2nd β€˜π‘§) ∈ π‘Œ) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑀 ∧ (2nd β€˜π‘§) ∈ π‘Œ)))
12 an32 644 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (1st β€˜π‘§) ∈ 𝑀) ∧ (2nd β€˜π‘§) ∈ π‘Œ) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (2nd β€˜π‘§) ∈ π‘Œ) ∧ (1st β€˜π‘§) ∈ 𝑀))
1310, 11, 123bitr2i 298 . . . . . . . . . . 11 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (2nd β€˜π‘§) ∈ π‘Œ) ∧ (1st β€˜π‘§) ∈ 𝑀))
1413baib 536 . . . . . . . . . 10 ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (2nd β€˜π‘§) ∈ π‘Œ) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (1st β€˜π‘§) ∈ 𝑀))
158, 9, 14syl2anc 584 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (1st β€˜π‘§) ∈ 𝑀))
167, 15bitr4d 281 . . . . . . . 8 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
1716pm5.32i 575 . . . . . . 7 ((𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
185, 17bitri 274 . . . . . 6 (𝑧 ∈ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
19 toponss 22420 . . . . . . . . . 10 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
2019adantlr 713 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
21 xpss1 5694 . . . . . . . . 9 (𝑀 βŠ† 𝑋 β†’ (𝑀 Γ— π‘Œ) βŠ† (𝑋 Γ— π‘Œ))
2220, 21syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) βŠ† (𝑋 Γ— π‘Œ))
2322sseld 3980 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) β†’ 𝑧 ∈ (𝑋 Γ— π‘Œ)))
2423pm4.71rd 563 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (𝑧 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ (𝑀 Γ— π‘Œ))))
2518, 24bitr4id 289 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑧 ∈ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ↔ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
2625eqrdv 2730 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) = (𝑀 Γ— π‘Œ))
27 toponmax 22419 . . . . . 6 (𝑆 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝑆)
2827ad2antlr 725 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ π‘Œ ∈ 𝑆)
29 txopn 23097 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑀 ∈ 𝑅 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3029anassrs 468 . . . . 5 ((((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) ∧ π‘Œ ∈ 𝑆) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3128, 30mpdan 685 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3226, 31eqeltrd 2833 . . 3 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))
3332ralrimiva 3146 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ βˆ€π‘€ ∈ 𝑅 (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))
34 txtopon 23086 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
35 simpl 483 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑅 ∈ (TopOnβ€˜π‘‹))
36 iscn 22730 . . 3 (((𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝑅 ∈ (TopOnβ€˜π‘‹)) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅) ↔ ((1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹ ∧ βˆ€π‘€ ∈ 𝑅 (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))))
3734, 35, 36syl2anc 584 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅) ↔ ((1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹ ∧ βˆ€π‘€ ∈ 𝑅 (β—‘(1st β†Ύ (𝑋 Γ— π‘Œ)) β€œ 𝑀) ∈ (𝑅 Γ—t 𝑆))))
382, 33, 37mpbir2and 711 1 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3947  βŸ¨cop 4633   Γ— cxp 5673  β—‘ccnv 5674   β†Ύ cres 5677   β€œ cima 5678   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  TopOnctopon 22403   Cn ccn 22719   Γ—t ctx 23055
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-tx 23057
This theorem is referenced by:  txcn  23121  txcmpb  23139  cnmpt1st  23163  sxbrsiga  33277  txsconnlem  34219  txsconn  34220  hausgraph  41939
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