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Theorem cnmptcom 23181
Description: The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
Hypotheses
Ref Expression
cnmptcom.3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmptcom.4 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmptcom.6 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Assertion
Ref Expression
cnmptcom (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
Distinct variable groups:   π‘₯,𝑦,𝐿   π‘₯,𝑋,𝑦   πœ‘,π‘₯,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)
Proof of Theorem cnmptcom
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmptcom.3 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 cnmptcom.4 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 txtopon 23094 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
41, 2, 3syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
5 cnmptcom.6 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
6 cntop2 22744 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
75, 6syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
8 toptopon2 22419 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
97, 8sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
10 cnf2 22752 . . . . . . . 8 (((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)
114, 9, 5, 10syl3anc 1371 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
12 eqid 2732 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
1312fmpo 8053 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
14 ralcom 3286 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 ↔ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
1513, 14bitr3i 276 . . . . . . 7 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿 ↔ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
1611, 15sylib 217 . . . . . 6 (πœ‘ β†’ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
17 eqid 2732 . . . . . . 7 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
1817fmpo 8053 . . . . . 6 (βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿 ↔ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
1916, 18sylib 217 . . . . 5 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
2019ffnd 6718 . . . 4 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) Fn (π‘Œ Γ— 𝑋))
21 fnov 7539 . . . 4 ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) Fn (π‘Œ Γ— 𝑋) ↔ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
2220, 21sylib 217 . . 3 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
23 nfcv 2903 . . . . . . 7 Ⅎ𝑦𝑧
24 nfcv 2903 . . . . . . 7 β„²π‘₯𝑧
25 nfcv 2903 . . . . . . 7 β„²π‘₯𝑀
26 nfv 1917 . . . . . . . 8 β„²π‘¦πœ‘
27 nfcv 2903 . . . . . . . . . 10 Ⅎ𝑦π‘₯
28 nfmpo2 7489 . . . . . . . . . 10 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
2927, 28, 23nfov 7438 . . . . . . . . 9 Ⅎ𝑦(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)
30 nfmpo1 7488 . . . . . . . . . 10 Ⅎ𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3123, 30, 27nfov 7438 . . . . . . . . 9 Ⅎ𝑦(𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)
3229, 31nfeq 2916 . . . . . . . 8 Ⅎ𝑦(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)
3326, 32nfim 1899 . . . . . . 7 Ⅎ𝑦(πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
34 nfv 1917 . . . . . . . 8 β„²π‘₯πœ‘
35 nfmpo1 7488 . . . . . . . . . 10 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
3625, 35, 24nfov 7438 . . . . . . . . 9 β„²π‘₯(𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)
37 nfmpo2 7489 . . . . . . . . . 10 β„²π‘₯(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3824, 37, 25nfov 7438 . . . . . . . . 9 β„²π‘₯(𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
3936, 38nfeq 2916 . . . . . . . 8 β„²π‘₯(𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
4034, 39nfim 1899 . . . . . . 7 β„²π‘₯(πœ‘ β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
41 oveq2 7416 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧))
42 oveq1 7415 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
4341, 42eqeq12d 2748 . . . . . . . 8 (𝑦 = 𝑧 β†’ ((π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) ↔ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
4443imbi2d 340 . . . . . . 7 (𝑦 = 𝑧 β†’ ((πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)) ↔ (πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))))
45 oveq1 7415 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧))
46 oveq2 7416 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
4745, 46eqeq12d 2748 . . . . . . . 8 (π‘₯ = 𝑀 β†’ ((π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) ↔ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
4847imbi2d 340 . . . . . . 7 (π‘₯ = 𝑀 β†’ ((πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)) ↔ (πœ‘ β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))))
49 rsp2 3274 . . . . . . . . 9 (βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿 β†’ ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐿))
5049, 16syl11 33 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (πœ‘ β†’ 𝐴 ∈ βˆͺ 𝐿))
5112ovmpt4g 7554 . . . . . . . . . . 11 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
52513com12 1123 . . . . . . . . . 10 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
5317ovmpt4g 7554 . . . . . . . . . 10 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
5452, 53eqtr4d 2775 . . . . . . . . 9 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
55543expia 1121 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (𝐴 ∈ βˆͺ 𝐿 β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
5650, 55syld 47 . . . . . . 7 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
5723, 24, 25, 33, 40, 44, 48, 56vtocl2gaf 3567 . . . . . 6 ((𝑧 ∈ π‘Œ ∧ 𝑀 ∈ 𝑋) β†’ (πœ‘ β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
5857com12 32 . . . . 5 (πœ‘ β†’ ((𝑧 ∈ π‘Œ ∧ 𝑀 ∈ 𝑋) β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
59583impib 1116 . . . 4 ((πœ‘ ∧ 𝑧 ∈ π‘Œ ∧ 𝑀 ∈ 𝑋) β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
6059mpoeq3dva 7485 . . 3 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)) = (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
6122, 60eqtr4d 2775 . 2 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)))
622, 1cnmpt2nd 23172 . . 3 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ 𝑀) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
632, 1cnmpt1st 23171 . . 3 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ 𝑧) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐾))
642, 1, 62, 63, 5cnmpt22f 23178 . 2 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
6561, 64eqeltrd 2833 1 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆͺ cuni 4908   Γ— cxp 5674   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Topctop 22394  TopOnctopon 22411   Cn ccn 22727   Γ—t ctx 23063
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cn 22730  df-tx 23065
This theorem is referenced by:  cnmpt2k  23191  htpycc  24495
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