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Theorem cnmpt2t 23168
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 6888 . . . . . . 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 7408 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)β€˜βŸ¨π‘’, π‘£βŸ©)
31, 2eqtr4di 2790 . . . . . 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 6888 . . . . . . 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 7408 . . . . . . 7 (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)β€˜βŸ¨π‘’, π‘£βŸ©)
64, 5eqtr4di 2790 . . . . . 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 4880 . . . . 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 7518 . . . 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 2903 . . . . . . 7 β„²π‘₯𝑒
10 nfmpo1 7485 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
11 nfcv 2903 . . . . . . 7 β„²π‘₯𝑣
129, 10, 11nfov 7435 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
13 nfmpo1 7485 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
149, 13, 11nfov 7435 . . . . . 6 β„²π‘₯(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
1512, 14nfop 4888 . . . . 5 β„²π‘₯⟨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
16 nfcv 2903 . . . . . . 7 Ⅎ𝑦𝑒
17 nfmpo2 7486 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
18 nfcv 2903 . . . . . . 7 Ⅎ𝑦𝑣
1916, 17, 18nfov 7435 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣)
20 nfmpo2 7486 . . . . . . 7 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
2116, 20, 18nfov 7435 . . . . . 6 Ⅎ𝑦(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)
2219, 21nfop 4888 . . . . 5 β„²π‘¦βŸ¨(𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣), (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣)⟩
23 nfcv 2903 . . . . 5 β„²π‘’βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
24 nfcv 2903 . . . . 5 β„²π‘£βŸ¨(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦), (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦)⟩
25 oveq12 7414 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦))
26 oveq12 7414 . . . . . 6 ((𝑒 = π‘₯ ∧ 𝑣 = 𝑦) β†’ (𝑒(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑣) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦))
2725, 26opeq12d 4880 . . . . 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 7499 . . . 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 2760 . . 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 23086 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
3330, 31, 32syl2anc 584 . . . 4 (πœ‘ β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
34 toponuni 22407 . . . 4 ((𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝐽 Γ—t 𝐾))
35 mpteq1 5240 . . . 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 1137 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
38 simp3 1138 . . . . . 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 22736 . . . . . . . . . . . 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 22411 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
4341, 42sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
44 cnf2 22744 . . . . . . . . . 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 1371 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
46 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
4746fmpo 8050 . . . . . . . . 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 3274 . . . . . . . 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 1116 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿)
5246ovmpt4g 7551 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1371 . . . . 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 22736 . . . . . . . . . . . 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 22411 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
5856, 57sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
59 cnf2 22744 . . . . . . . . . 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 1371 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝑀)
61 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
6261fmpo 8050 . . . . . . . . 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 3274 . . . . . . . 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 1116 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ βˆͺ 𝑀)
6761ovmpt4g 7551 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐡 ∈ βˆͺ 𝑀) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6837, 38, 66, 67syl3anc 1371 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)𝑦) = 𝐡)
6953, 68opeq12d 4880 . . . 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 7482 . . 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 2796 . 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 2732 . . . 4 βˆͺ (𝐽 Γ—t 𝐾) = βˆͺ (𝐽 Γ—t 𝐾)
73 eqid 2732 . . . 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 23119 . . 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 584 . 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 2834 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸ¨cop 4633  βˆͺ cuni 4907   ↦ cmpt 5230   Γ— cxp 5673  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Topctop 22386  TopOnctopon 22403   Cn ccn 22719   Γ—t ctx 23055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-tx 23057
This theorem is referenced by:  cnmpt22  23169  txhmeo  23298  txswaphmeo  23300  txsconnlem  34219
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