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Theorem tx1cn 21692
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 7390 . . 3 (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3 toponss 21011 . . . . . . . . . 10 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤𝑅) → 𝑤𝑋)
43adantlr 706 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑤𝑋)
5 xpss1 5296 . . . . . . . . 9 (𝑤𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
64, 5syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
76sseld 3760 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
87pm4.71rd 558 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
9 ffn 6223 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
10 elpreima 6527 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
111, 9, 10mp2b 10 . . . . . . 7 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
12 fvres 6394 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
1312eleq1d 2829 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1st𝑧) ∈ 𝑤))
14 1st2nd2 7405 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
15 xp2nd 7399 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
16 elxp6 7400 . . . . . . . . . . . 12 (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
17 anass 460 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
18 an32 636 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1916, 17, 183bitr2i 290 . . . . . . . . . . 11 (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
2019baib 531 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
2114, 15, 20syl2anc 579 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
2213, 21bitr4d 273 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑤 × 𝑌)))
2322pm5.32i 570 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
2411, 23bitri 266 . . . . . 6 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
258, 24syl6rbbr 281 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
2625eqrdv 2763 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27 toponmax 21010 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2827ad2antlr 718 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑌𝑆)
29 txopn 21685 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤𝑅𝑌𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3029anassrs 459 . . . . 5 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) ∧ 𝑌𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3128, 30mpdan 678 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3226, 31eqeltrd 2844 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3332ralrimiva 3113 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
34 txtopon 21674 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35 simpl 474 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36 iscn 21319 . . 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 579 . 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 704 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732  cop 4340   × cxp 5275  ccnv 5276  cres 5279  cima 5280   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  TopOnctopon 20994   Cn ccn 21308   ×t ctx 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-tx 21645
This theorem is referenced by:  txcn  21709  txcmpb  21727  cnmpt1st  21751  sxbrsiga  30734  txsconnlem  31602  txsconn  31603  hausgraph  38399
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