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Theorem tx2cn 21394
Description: Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx2cn ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Proof of Theorem tx2cn
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f2ndres 7176 . . 3 (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌)
3 toponss 20712 . . . . . . . . . 10 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤𝑆) → 𝑤𝑌)
43adantll 749 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → 𝑤𝑌)
5 xpss2 5219 . . . . . . . . 9 (𝑤𝑌 → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
64, 5syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
76sseld 3594 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
87pm4.71rd 666 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
9 ffn 6032 . . . . . . . 8 ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
10 elpreima 6323 . . . . . . . 8 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
111, 9, 10mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
12 fvres 6194 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
1312eleq1d 2684 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (2nd𝑧) ∈ 𝑤))
14 1st2nd2 7190 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
15 xp1st 7183 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
16 elxp6 7185 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
17 anass 680 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
1816, 17bitr4i 267 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤))
1918baib 943 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
2014, 15, 19syl2anc 692 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
2113, 20bitr4d 271 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑋 × 𝑤)))
2221pm5.32i 668 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
2311, 22bitri 264 . . . . . 6 (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
248, 23syl6rbbr 279 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑋 × 𝑤)))
2524eqrdv 2618 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑋 × 𝑤))
26 toponmax 20711 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
27 txopn 21386 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑋𝑅𝑤𝑆)) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
2827expr 642 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑋𝑅) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
2926, 28mpidan 703 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
3029imp 445 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
3125, 30eqeltrd 2699 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3231ralrimiva 2963 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33 txtopon 21375 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34 iscn 21020 . . 3 (((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
3533, 34sylancom 700 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
362, 32, 35mpbir2and 956 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wss 3567  cop 4174   × cxp 5102  ccnv 5103  cres 5106  cima 5107   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  TopOnctopon 20696   Cn ccn 21009   ×t ctx 21344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cn 21012  df-tx 21346
This theorem is referenced by:  txcn  21410  txcmpb  21428  txkgen  21436  cnmpt2nd  21453  sxbrsiga  30326  txsconnlem  31196  txsconn  31197  hausgraph  37609
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