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Theorem tx1cn 14437
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 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))

Proof of Theorem tx1cn
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 6212 . . 3 (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
21a1i 9 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3 ffn 5403 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4 elpreima 5677 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
51, 3, 4mp2b 8 . . . . . . 7 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6 fvres 5578 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
76eleq1d 2262 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1st𝑧) ∈ 𝑤))
8 1st2nd2 6228 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
9 xp2nd 6219 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
10 elxp6 6222 . . . . . . . . . . . 12 (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
11 anass 401 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
12 an32 562 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1310, 11, 123bitr2i 208 . . . . . . . . . . 11 (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1413baib 920 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
158, 9, 14syl2anc 411 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
167, 15bitr4d 191 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑤 × 𝑌)))
1716pm5.32i 454 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
185, 17bitri 184 . . . . . 6 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19 toponss 14194 . . . . . . . . . 10 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤𝑅) → 𝑤𝑋)
2019adantlr 477 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑤𝑋)
21 xpss1 4769 . . . . . . . . 9 (𝑤𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
2220, 21syl 14 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
2322sseld 3178 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
2423pm4.71rd 394 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
2518, 24bitr4id 199 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
2625eqrdv 2191 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27 toponmax 14193 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2827ad2antlr 489 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑌𝑆)
29 txopn 14433 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤𝑅𝑌𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3029anassrs 400 . . . . 5 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) ∧ 𝑌𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3128, 30mpdan 421 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3226, 31eqeltrd 2270 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3332ralrimiva 2567 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
34 txtopon 14430 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35 simpl 109 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36 iscn 14365 . . 3 (((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑅 ∈ (TopOn‘𝑋)) → ((1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ↔ ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
3734, 35, 36syl2anc 411 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ↔ ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
382, 33, 37mpbir2and 946 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  wral 2472  wss 3153  cop 3621   × cxp 4657  ccnv 4658  cres 4661  cima 4662   Fn wfn 5249  wf 5250  cfv 5254  (class class class)co 5918  1st c1st 6191  2nd c2nd 6192  TopOnctopon 14178   Cn ccn 14353   ×t ctx 14420
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-bases 14211  df-cn 14356  df-tx 14421
This theorem is referenced by:  txcn  14443  cnmpt1st  14456
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