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Theorem tx1cn 13262
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 6150 . . 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 5357 . . . . . . . 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 5627 . . . . . . . 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 5531 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
76eleq1d 2244 . . . . . . . . 9 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) ∈ 𝑀 ↔ (1st β€˜π‘§) ∈ 𝑀))
8 1st2nd2 6166 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
9 xp2nd 6157 . . . . . . . . . 10 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘§) ∈ π‘Œ)
10 elxp6 6160 . . . . . . . . . . . 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 13017 . . . . . . . . . 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 4730 . . . . . . . . 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 3152 . . . . . . 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 2173 . . . 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 13016 . . . . . 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 13258 . . . . . 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 2252 . . 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 2548 . 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 13255 . . 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 13190 . . 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 2146  βˆ€wral 2453   βŠ† wss 3127  βŸ¨cop 3592   Γ— cxp 4618  β—‘ccnv 4619   β†Ύ cres 4622   β€œ cima 4623   Fn wfn 5203  βŸΆwf 5204  β€˜cfv 5208  (class class class)co 5865  1st c1st 6129  2nd c2nd 6130  TopOnctopon 13001   Cn ccn 13178   Γ—t ctx 13245
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-topgen 12630  df-top 12989  df-topon 13002  df-bases 13034  df-cn 13181  df-tx 13246
This theorem is referenced by:  txcn  13268  cnmpt1st  13281
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