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Theorem cnmpt2t 23177
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 6892 . . . . . . 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 7412 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜βŸ¨π‘’, π‘£βŸ©)
31, 2eqtr4di 2791 . . . . . 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 6892 . . . . . . 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 7412 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜βŸ¨π‘’, π‘£βŸ©)
64, 5eqtr4di 2791 . . . . . 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 4882 . . . . 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 7522 . . . 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 2904 . . . . . . 7 β„²π‘₯𝑒
10 nfmpo1 7489 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
11 nfcv 2904 . . . . . . 7 β„²π‘₯𝑣
129, 10, 11nfov 7439 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
13 nfmpo1 7489 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
149, 13, 11nfov 7439 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
1512, 14nfop 4890 . . . . 5 β„²π‘₯⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
16 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝑒
17 nfmpo2 7490 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
18 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝑣
1916, 17, 18nfov 7439 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
20 nfmpo2 7490 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
2116, 20, 18nfov 7439 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
2219, 21nfop 4890 . . . . 5 β„²π‘¦βŸ¨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
23 nfcv 2904 . . . . 5 β„²π‘’βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
24 nfcv 2904 . . . . 5 β„²π‘£βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
25 oveq12 7418 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦))
26 oveq12 7418 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦))
2725, 26opeq12d 4882 . . . . 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 7503 . . . 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 2761 . . 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 23095 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
3330, 31, 32syl2anc 585 . . . 4 (πœ‘ β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
34 toponuni 22416 . . . 4 ((𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝐽 Γ—t 𝐾))
35 mpteq1 5242 . . . 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 18 . . 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 1138 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
38 simp3 1139 . . . . . 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 22745 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
4139, 40syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ Top)
42 toptopon2 22420 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
4341, 42sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
44 cnf2 22753 . . . . . . . . . 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 1372 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
46 eqid 2733 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
4746fmpo 8054 . . . . . . . . 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 233 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿)
49 rsp2 3275 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿))
5048, 49syl 17 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿))
51503impib 1117 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿)
5246ovmpt4g 7555 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1372 . . . . 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 22745 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀) β†’ 𝑀 ∈ Top)
5654, 55syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑀 ∈ Top)
57 toptopon2 22420 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
5856, 57sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
59 cnf2 22753 . . . . . . . . . 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 1372 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝑀)
61 eqid 2733 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
6261fmpo 8054 . . . . . . . . 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 233 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝑀)
64 rsp2 3275 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝑀 β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀))
6563, 64syl 17 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀))
66653impib 1117 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀)
6761ovmpt4g 7555 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐡 ∈ βˆͺ 𝑀) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6837, 38, 66, 67syl3anc 1372 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6953, 68opeq12d 4882 . . . 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 7486 . . 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 2797 . 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 2733 . . . 4 βˆͺ (𝐽 Γ—t 𝐾) = βˆͺ (𝐽 Γ—t 𝐾)
73 eqid 2733 . . . 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 23128 . . 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 585 . 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 2835 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸ¨cop 4635  βˆͺ cuni 4909   ↦ cmpt 5232   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Topctop 22395  TopOnctopon 22412   Cn ccn 22728   Γ—t ctx 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cn 22731  df-tx 23066
This theorem is referenced by:  cnmpt22  23178  txhmeo  23307  txswaphmeo  23309  txsconnlem  34262
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