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Theorem tx1cn 23457
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 7993 . . 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 6708 . . . . . . . 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 7050 . . . . . . . 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 6901 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
76eleq1d 2810 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (1st β€˜π‘§) ∈ 𝑀))
8 1st2nd2 8008 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp2nd 8002 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘§) ∈ π‘Œ)
10 elxp6 8003 . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 22773 . . . . . . . . . 10 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
2019adantlr 712 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
21 xpss1 5686 . . . . . . . . 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 3974 . . . . . . 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 2722 . . . 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 22772 . . . . . 6 (𝑆 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝑆)
2827ad2antlr 724 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ π‘Œ ∈ 𝑆)
29 txopn 23450 . . . . . 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 684 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3226, 31eqeltrd 2825 . . 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 3138 . 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 23439 . . 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 23083 . . 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 710 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 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  βŸ¨cop 4627   Γ— cxp 5665  β—‘ccnv 5666   β†Ύ cres 5669   β€œ cima 5670   Fn wfn 6529  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  TopOnctopon 22756   Cn ccn 23072   Γ—t ctx 23408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cn 23075  df-tx 23410
This theorem is referenced by:  txcn  23474  txcmpb  23492  cnmpt1st  23516  sxbrsiga  33809  txsconnlem  34749  txsconn  34750  hausgraph  42504
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