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Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnmpt2t GIF version
Theorem cnmpt2t 13796
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt21.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt21.a (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
cnmpt2t.b (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
Assertion
Ref Expression
cnmpt2t (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
Distinct variable groups:   π‘₯,𝑦,𝐿   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑀,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)
Proof of Theorem cnmpt2t
Dummy variables 𝑣 𝑒 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5516 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜βŸ¨π‘’, π‘£βŸ©))
2 df-ov 5878 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜βŸ¨π‘’, π‘£βŸ©)
31, 2eqtr4di 2228 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§) = (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣))
4 fveq2 5516 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜βŸ¨π‘’, π‘£βŸ©))
5 df-ov 5878 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜βŸ¨π‘’, π‘£βŸ©)
64, 5eqtr4di 2228 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§) = (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣))
73, 6opeq12d 3787 . . . . 5 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩ = ⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩)
87mpompt 5967 . . . 4 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (𝑒 ∈ 𝑋, 𝑣 ∈ π‘Œ ↦ ⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩)
9 nfcv 2319 . . . . . . 7 β„²π‘₯𝑒
10 nfmpo1 5942 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
11 nfcv 2319 . . . . . . 7 β„²π‘₯𝑣
129, 10, 11nfov 5905 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
13 nfmpo1 5942 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
149, 13, 11nfov 5905 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
1512, 14nfop 3795 . . . . 5 β„²π‘₯⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
16 nfcv 2319 . . . . . . 7 Ⅎ𝑦𝑒
17 nfmpo2 5943 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
18 nfcv 2319 . . . . . . 7 Ⅎ𝑦𝑣
1916, 17, 18nfov 5905 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
20 nfmpo2 5943 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
2116, 20, 18nfov 5905 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
2219, 21nfop 3795 . . . . 5 β„²π‘¦βŸ¨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
23 nfcv 2319 . . . . 5 β„²π‘’βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
24 nfcv 2319 . . . . 5 β„²π‘£βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
25 oveq12 5884 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦))
26 oveq12 5884 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦))
2725, 26opeq12d 3787 . . . . 5 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ ⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩ = ⟨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩)
2815, 22, 23, 24, 27cbvmpo 5954 . . . 4 (𝑒 ∈ 𝑋, 𝑣 ∈ π‘Œ ↦ ⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩)
298, 28eqtri 2198 . . 3 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩)
30 cnmpt21.j . . . . 5 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
31 cnmpt21.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
32 txtopon 13765 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
3330, 31, 32syl2anc 411 . . . 4 (πœ‘ β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
34 toponuni 13518 . . . 4 ((𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝐽 Γ—t 𝐾))
35 mpteq1 4088 . . . 4 ((𝑋 Γ— π‘Œ) = βˆͺ (𝐽 Γ—t 𝐾) β†’ (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩))
3633, 34, 353syl 17 . . 3 (πœ‘ β†’ (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩))
37 simp2 998 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
38 simp3 999 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
39 cnmpt21.a . . . . . . . . . . . 12 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
40 cntop2 13705 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
4139, 40syl 14 . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ Top)
42 toptopon2 13522 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
4341, 42sylib 122 . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
44 cnf2 13708 . . . . . . . . . 10 (((𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
4533, 43, 39, 44syl3anc 1238 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
46 eqid 2177 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
4746fmpo 6202 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
4845, 47sylibr 134 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿)
49 rsp2 2527 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿))
5048, 49syl 14 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿))
51503impib 1201 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿)
5246ovmpt4g 5997 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1238 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
54 cnmpt2t.b . . . . . . . . . . . 12 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
55 cntop2 13705 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀) β†’ 𝑀 ∈ Top)
5654, 55syl 14 . . . . . . . . . . 11 (πœ‘ β†’ 𝑀 ∈ Top)
57 toptopon2 13522 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
5856, 57sylib 122 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
59 cnf2 13708 . . . . . . . . . 10 (((𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀) ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝑀)
6033, 58, 54, 59syl3anc 1238 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝑀)
61 eqid 2177 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
6261fmpo 6202 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝑀 ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝑀)
6360, 62sylibr 134 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝑀)
64 rsp2 2527 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝑀 β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀))
6563, 64syl 14 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀))
66653impib 1201 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀)
6761ovmpt4g 5997 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐡 ∈ βˆͺ 𝑀) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6837, 38, 66, 67syl3anc 1238 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6953, 68opeq12d 3787 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ ⟨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩ = ⟨𝐴, 𝐡⟩)
7069mpoeq3dva 5939 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩))
7129, 36, 703eqtr3a 2234 . 2 (πœ‘ β†’ (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩))
72 eqid 2177 . . . 4 βˆͺ (𝐽 Γ—t 𝐾) = βˆͺ (𝐽 Γ—t 𝐾)
73 eqid 2177 . . . 4 (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) = (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩)
7472, 73txcnmpt 13776 . . 3 (((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀)) β†’ (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
7539, 54, 74syl2anc 411 . 2 (πœ‘ β†’ (𝑧 ∈ βˆͺ (𝐽 Γ—t 𝐾) ↦ ⟨((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§), ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π‘§)⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
7671, 75eqeltrrd 2255 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βŸ¨cop 3596  βˆͺ cuni 3810   ↦ cmpt 4065   Γ— cxp 4625  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875   ∈ cmpo 5877  Topctop 13500  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-nul 3424  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:  cnmpt22  13797  txhmeo  13822  txswaphmeo  13824
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