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Theorem tx1cn 13772
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 6160 . . 3 (1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹
21a1i 9 . 2 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)):(𝑋 Γ— π‘Œ)βŸΆπ‘‹)
3 ffn 5366 . . . . . . . 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 5636 . . . . . . . 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 8 . . . . . . 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 5540 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
76eleq1d 2246 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (1st β€˜π‘§) ∈ 𝑀))
8 1st2nd2 6176 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp2nd 6167 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘§) ∈ π‘Œ)
10 elxp6 6170 . . . . . . . . . . . 12 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ ((1st β€˜π‘§) ∈ 𝑀 ∧ (2nd β€˜π‘§) ∈ π‘Œ)))
11 anass 401 . . . . . . . . . . . 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 562 . . . . . . . . . . . 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 208 . . . . . . . . . . 11 (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ ((𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∧ (2nd β€˜π‘§) ∈ π‘Œ) ∧ (1st β€˜π‘§) ∈ 𝑀))
1413baib 919 . . . . . . . . . 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 411 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) ↔ (1st β€˜π‘§) ∈ 𝑀))
167, 15bitr4d 191 . . . . . . . 8 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ 𝑧 ∈ (𝑀 Γ— π‘Œ)))
1716pm5.32i 454 . . . . . . 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 184 . . . . . 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 13529 . . . . . . . . . 10 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
2019adantlr 477 . . . . . . . . 9 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ 𝑀 βŠ† 𝑋)
21 xpss1 4737 . . . . . . . . 9 (𝑀 βŠ† 𝑋 β†’ (𝑀 Γ— π‘Œ) βŠ† (𝑋 Γ— π‘Œ))
2220, 21syl 14 . . . . . . . 8 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) βŠ† (𝑋 Γ— π‘Œ))
2322sseld 3155 . . . . . . 7 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑧 ∈ (𝑀 Γ— π‘Œ) β†’ 𝑧 ∈ (𝑋 Γ— π‘Œ)))
2423pm4.71rd 394 . . . . . 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 199 . . . . 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 2175 . . . 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 13528 . . . . . 6 (𝑆 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝑆)
2827ad2antlr 489 . . . . 5 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ π‘Œ ∈ 𝑆)
29 txopn 13768 . . . . . 6 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑀 ∈ 𝑅 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3029anassrs 400 . . . . 5 ((((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) ∧ π‘Œ ∈ 𝑆) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3128, 30mpdan 421 . . . 4 (((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑀 ∈ 𝑅) β†’ (𝑀 Γ— π‘Œ) ∈ (𝑅 Γ—t 𝑆))
3226, 31eqeltrd 2254 . . 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 2550 . 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 13765 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑅 Γ—t 𝑆) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
35 simpl 109 . . 3 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑅 ∈ (TopOnβ€˜π‘‹))
36 iscn 13700 . . 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 411 . 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 944 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455   βŠ† wss 3130  βŸ¨cop 3596   Γ— cxp 4625  β—‘ccnv 4626   β†Ύ cres 4629   β€œ cima 4630   Fn wfn 5212  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875  1st c1st 6139  2nd c2nd 6140  TopOnctopon 13513   Cn ccn 13688   Γ—t ctx 13755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-topgen 12709  df-top 13501  df-topon 13514  df-bases 13546  df-cn 13691  df-tx 13756
This theorem is referenced by:  txcn  13778  cnmpt1st  13791
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