ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnmptcom GIF version
Theorem cnmptcom 13837
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 13801 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
41, 2, 3syl2anc 411 . . . . . . . 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 13741 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
75, 6syl 14 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
8 toptopon2 13558 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
97, 8sylib 122 . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
10 cnf2 13744 . . . . . . . 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 1238 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢βˆͺ 𝐿)
12 eqid 2177 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
1312fmpo 6204 . . . . . . . 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 2640 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ βˆͺ 𝐿 ↔ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
1513, 14bitr3i 186 . . . . . . 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 122 . . . . . 6 (πœ‘ β†’ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
17 eqid 2177 . . . . . . 7 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
1817fmpo 6204 . . . . . 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 122 . . . . 5 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
2019ffnd 5368 . . . 4 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) Fn (π‘Œ Γ— 𝑋))
21 fnovim 5985 . . . 4 ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) Fn (π‘Œ Γ— 𝑋) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)))
2220, 21syl 14 . . 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 2319 . . . . . . 7 Ⅎ𝑦𝑧
24 nfcv 2319 . . . . . . 7 β„²π‘₯𝑧
25 nfcv 2319 . . . . . . 7 β„²π‘₯𝑀
26 nfv 1528 . . . . . . . 8 β„²π‘¦πœ‘
27 nfcv 2319 . . . . . . . . . 10 Ⅎ𝑦π‘₯
28 nfmpo2 5945 . . . . . . . . . 10 Ⅎ𝑦(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
2927, 28, 23nfov 5907 . . . . . . . . 9 Ⅎ𝑦(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)
30 nfmpo1 5944 . . . . . . . . . 10 Ⅎ𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3123, 30, 27nfov 5907 . . . . . . . . 9 Ⅎ𝑦(𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)
3229, 31nfeq 2327 . . . . . . . 8 Ⅎ𝑦(π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)
3326, 32nfim 1572 . . . . . . 7 Ⅎ𝑦(πœ‘ β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
34 nfv 1528 . . . . . . . 8 β„²π‘₯πœ‘
35 nfmpo1 5944 . . . . . . . . . 10 β„²π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
3625, 35, 24nfov 5907 . . . . . . . . 9 β„²π‘₯(𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧)
37 nfmpo2 5945 . . . . . . . . . 10 β„²π‘₯(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3824, 37, 25nfov 5907 . . . . . . . . 9 β„²π‘₯(𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
3936, 38nfeq 2327 . . . . . . . 8 β„²π‘₯(𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
4034, 39nfim 1572 . . . . . . 7 β„²π‘₯(πœ‘ β†’ (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
41 oveq2 5885 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧))
42 oveq1 5884 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
4341, 42eqeq12d 2192 . . . . . . . 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 230 . . . . . . 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 5884 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧) = (𝑀(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑧))
46 oveq2 5885 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = (𝑧(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
4745, 46eqeq12d 2192 . . . . . . . 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 230 . . . . . . 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 2527 . . . . . . . . 9 (βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿 β†’ ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐿))
5049, 16syl11 31 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (πœ‘ β†’ 𝐴 ∈ βˆͺ 𝐿))
5112ovmpt4g 5999 . . . . . . . . . . 11 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
52513com12 1207 . . . . . . . . . 10 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (π‘₯(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)𝑦) = 𝐴)
5317ovmpt4g 5999 . . . . . . . . . 10 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
5452, 53eqtr4d 2213 . . . . . . . . 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 1205 . . . . . . . 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 45 . . . . . . 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 2806 . . . . . 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 30 . . . . 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 1201 . . . 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 5941 . . 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 2213 . 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 13828 . . 3 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ 𝑀) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
632, 1cnmpt1st 13827 . . 3 (πœ‘ β†’ (𝑧 ∈ π‘Œ, 𝑀 ∈ 𝑋 ↦ 𝑧) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐾))
642, 1, 62, 63, 5cnmpt22f 13834 . 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 2254 1 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆͺ cuni 3811   Γ— cxp 4626   Fn wfn 5213  βŸΆwf 5214  β€˜cfv 5218  (class class class)co 5877   ∈ cmpo 5879  Topctop 13536  TopOnctopon 13549   Cn ccn 13724   Γ—t ctx 13791
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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-topgen 12714  df-top 13537  df-topon 13550  df-bases 13582  df-cn 13727  df-tx 13792
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator