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Theorem tx1cn 23526
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 8017 . . 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 6722 . . . . . . . 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 7067 . . . . . . . 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 6916 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
76eleq1d 2814 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (1st β€˜π‘§) ∈ 𝑀))
8 1st2nd2 8032 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp2nd 8026 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘§) ∈ π‘Œ)
10 elxp6 8027 . . . . . . . . . . . 12 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑀 ∧ (2nd β€˜π‘§) ∈ π‘Œ)))
11 anass 468 . . . . . . . . . . . 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 645 . . . . . . . . . . . 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 299 . . . . . . . . . . 11 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (2nd β€˜π‘§) ∈ π‘Œ) ∧ (1st β€˜π‘§) ∈ 𝑀))
1413baib 535 . . . . . . . . . 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 583 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (1st β€˜π‘§) ∈ 𝑀))
167, 15bitr4d 282 . . . . . . . 8 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
1716pm5.32i 574 . . . . . . 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 275 . . . . . 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 22842 . . . . . . . . . 10 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
2019adantlr 714 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
21 xpss1 5697 . . . . . . . . 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 3979 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) β†’ 𝑧 ∈ (𝑋 Γ— π‘Œ)))
2423pm4.71rd 562 . . . . . 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 290 . . . . 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 2726 . . . 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 22841 . . . . . 6 (𝑆 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝑆)
2827ad2antlr 726 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ π‘Œ ∈ 𝑆)
29 txopn 23519 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑀 ∈ 𝑅 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3029anassrs 467 . . . . 5 ((((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) ∧ π‘Œ ∈ 𝑆) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3128, 30mpdan 686 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3226, 31eqeltrd 2829 . . 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 3143 . 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 23508 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
35 simpl 482 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑅 ∈ (TopOnβ€˜π‘‹))
36 iscn 23152 . . 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 583 . 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 712 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 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βŠ† wss 3947  βŸ¨cop 4635   Γ— cxp 5676  β—‘ccnv 5677   β†Ύ cres 5680   β€œ cima 5681   Fn wfn 6543  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  TopOnctopon 22825   Cn ccn 23141   Γ—t ctx 23477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cn 23144  df-tx 23479
This theorem is referenced by:  txcn  23543  txcmpb  23561  cnmpt1st  23585  sxbrsiga  33910  txsconnlem  34850  txsconn  34851  hausgraph  42633
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