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Theorem ptcmpfi 23309
Description: A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
Assertion
Ref Expression
ptcmpfi ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜πΉ) ∈ Comp)
Proof of Theorem ptcmpfi
Dummy variables 𝑣 𝑒 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6715 . . . . 5 (𝐹:𝐴⟢Comp β†’ 𝐹 Fn 𝐴)
2 fnresdm 6667 . . . . 5 (𝐹 Fn 𝐴 β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
31, 2syl 17 . . . 4 (𝐹:𝐴⟢Comp β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
43adantl 483 . . 3 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
54fveq2d 6893 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) = (∏tβ€˜πΉ))
6 ssid 4004 . . . 4 𝐴 βŠ† 𝐴
7 sseq1 4007 . . . . . 6 (π‘₯ = βˆ… β†’ (π‘₯ βŠ† 𝐴 ↔ βˆ… βŠ† 𝐴))
8 reseq2 5975 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ (𝐹 β†Ύ π‘₯) = (𝐹 β†Ύ βˆ…))
9 res0 5984 . . . . . . . . . 10 (𝐹 β†Ύ βˆ…) = βˆ…
108, 9eqtrdi 2789 . . . . . . . . 9 (π‘₯ = βˆ… β†’ (𝐹 β†Ύ π‘₯) = βˆ…)
1110fveq2d 6893 . . . . . . . 8 (π‘₯ = βˆ… β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) = (∏tβ€˜βˆ…))
1211eleq1d 2819 . . . . . . 7 (π‘₯ = βˆ… β†’ ((∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp ↔ (∏tβ€˜βˆ…) ∈ Comp))
1312imbi2d 341 . . . . . 6 (π‘₯ = βˆ… β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜βˆ…) ∈ Comp)))
147, 13imbi12d 345 . . . . 5 (π‘₯ = βˆ… β†’ ((π‘₯ βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp)) ↔ (βˆ… βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜βˆ…) ∈ Comp))))
15 sseq1 4007 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝑦 βŠ† 𝐴))
16 reseq2 5975 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (𝐹 β†Ύ π‘₯) = (𝐹 β†Ύ 𝑦))
1716fveq2d 6893 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) = (∏tβ€˜(𝐹 β†Ύ 𝑦)))
1817eleq1d 2819 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp ↔ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp))
1918imbi2d 341 . . . . . 6 (π‘₯ = 𝑦 β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)))
2015, 19imbi12d 345 . . . . 5 (π‘₯ = 𝑦 β†’ ((π‘₯ βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp)) ↔ (𝑦 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp))))
21 sseq1 4007 . . . . . 6 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴))
22 reseq2 5975 . . . . . . . . 9 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ (𝐹 β†Ύ π‘₯) = (𝐹 β†Ύ (𝑦 βˆͺ {𝑧})))
2322fveq2d 6893 . . . . . . . 8 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) = (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))))
2423eleq1d 2819 . . . . . . 7 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ ((∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp ↔ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))
2524imbi2d 341 . . . . . 6 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp)))
2621, 25imbi12d 345 . . . . 5 (π‘₯ = (𝑦 βˆͺ {𝑧}) β†’ ((π‘₯ βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp)) ↔ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))))
27 sseq1 4007 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
28 reseq2 5975 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ (𝐹 β†Ύ π‘₯) = (𝐹 β†Ύ 𝐴))
2928fveq2d 6893 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) = (∏tβ€˜(𝐹 β†Ύ 𝐴)))
3029eleq1d 2819 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp ↔ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp))
3130imbi2d 341 . . . . . 6 (π‘₯ = 𝐴 β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp)))
3227, 31imbi12d 345 . . . . 5 (π‘₯ = 𝐴 β†’ ((π‘₯ βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ π‘₯)) ∈ Comp)) ↔ (𝐴 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp))))
33 0ex 5307 . . . . . . . . 9 βˆ… ∈ V
34 f0 6770 . . . . . . . . 9 βˆ…:βˆ…βŸΆTop
35 pttop 23078 . . . . . . . . 9 ((βˆ… ∈ V ∧ βˆ…:βˆ…βŸΆTop) β†’ (∏tβ€˜βˆ…) ∈ Top)
3633, 34, 35mp2an 691 . . . . . . . 8 (∏tβ€˜βˆ…) ∈ Top
37 eqid 2733 . . . . . . . . . . . . 13 (∏tβ€˜βˆ…) = (∏tβ€˜βˆ…)
3837ptuni 23090 . . . . . . . . . . . 12 ((βˆ… ∈ V ∧ βˆ…:βˆ…βŸΆTop) β†’ Xπ‘₯ ∈ βˆ… βˆͺ (βˆ…β€˜π‘₯) = βˆͺ (∏tβ€˜βˆ…))
3933, 34, 38mp2an 691 . . . . . . . . . . 11 Xπ‘₯ ∈ βˆ… βˆͺ (βˆ…β€˜π‘₯) = βˆͺ (∏tβ€˜βˆ…)
40 ixp0x 8917 . . . . . . . . . . . 12 Xπ‘₯ ∈ βˆ… βˆͺ (βˆ…β€˜π‘₯) = {βˆ…}
41 snfi 9041 . . . . . . . . . . . 12 {βˆ…} ∈ Fin
4240, 41eqeltri 2830 . . . . . . . . . . 11 Xπ‘₯ ∈ βˆ… βˆͺ (βˆ…β€˜π‘₯) ∈ Fin
4339, 42eqeltrri 2831 . . . . . . . . . 10 βˆͺ (∏tβ€˜βˆ…) ∈ Fin
44 pwfi 9175 . . . . . . . . . 10 (βˆͺ (∏tβ€˜βˆ…) ∈ Fin ↔ 𝒫 βˆͺ (∏tβ€˜βˆ…) ∈ Fin)
4543, 44mpbi 229 . . . . . . . . 9 𝒫 βˆͺ (∏tβ€˜βˆ…) ∈ Fin
46 pwuni 4949 . . . . . . . . 9 (∏tβ€˜βˆ…) βŠ† 𝒫 βˆͺ (∏tβ€˜βˆ…)
47 ssfi 9170 . . . . . . . . 9 ((𝒫 βˆͺ (∏tβ€˜βˆ…) ∈ Fin ∧ (∏tβ€˜βˆ…) βŠ† 𝒫 βˆͺ (∏tβ€˜βˆ…)) β†’ (∏tβ€˜βˆ…) ∈ Fin)
4845, 46, 47mp2an 691 . . . . . . . 8 (∏tβ€˜βˆ…) ∈ Fin
4936, 48elini 4193 . . . . . . 7 (∏tβ€˜βˆ…) ∈ (Top ∩ Fin)
50 fincmp 22889 . . . . . . 7 ((∏tβ€˜βˆ…) ∈ (Top ∩ Fin) β†’ (∏tβ€˜βˆ…) ∈ Comp)
5149, 50ax-mp 5 . . . . . 6 (∏tβ€˜βˆ…) ∈ Comp
52512a1i 12 . . . . 5 (βˆ… βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜βˆ…) ∈ Comp))
53 ssun1 4172 . . . . . . . . 9 𝑦 βŠ† (𝑦 βˆͺ {𝑧})
54 id 22 . . . . . . . . 9 ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)
5553, 54sstrid 3993 . . . . . . . 8 ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ 𝑦 βŠ† 𝐴)
5655imim1i 63 . . . . . . 7 ((𝑦 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)) β†’ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)))
57 eqid 2733 . . . . . . . . . . . . . 14 βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦)) = βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦))
58 eqid 2733 . . . . . . . . . . . . . 14 βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧})) = βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧}))
59 eqid 2733 . . . . . . . . . . . . . 14 (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) = (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧})))
60 resabs1 6010 . . . . . . . . . . . . . . . . 17 (𝑦 βŠ† (𝑦 βˆͺ {𝑧}) β†’ ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ 𝑦) = (𝐹 β†Ύ 𝑦))
6153, 60ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ 𝑦) = (𝐹 β†Ύ 𝑦)
6261eqcomi 2742 . . . . . . . . . . . . . . 15 (𝐹 β†Ύ 𝑦) = ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ 𝑦)
6362fveq2i 6892 . . . . . . . . . . . . . 14 (∏tβ€˜(𝐹 β†Ύ 𝑦)) = (∏tβ€˜((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ 𝑦))
64 ssun2 4173 . . . . . . . . . . . . . . . . 17 {𝑧} βŠ† (𝑦 βˆͺ {𝑧})
65 resabs1 6010 . . . . . . . . . . . . . . . . 17 ({𝑧} βŠ† (𝑦 βˆͺ {𝑧}) β†’ ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ {𝑧}) = (𝐹 β†Ύ {𝑧}))
6664, 65ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ {𝑧}) = (𝐹 β†Ύ {𝑧})
6766eqcomi 2742 . . . . . . . . . . . . . . 15 (𝐹 β†Ύ {𝑧}) = ((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ {𝑧})
6867fveq2i 6892 . . . . . . . . . . . . . 14 (∏tβ€˜(𝐹 β†Ύ {𝑧})) = (∏tβ€˜((𝐹 β†Ύ (𝑦 βˆͺ {𝑧})) β†Ύ {𝑧}))
69 eqid 2733 . . . . . . . . . . . . . 14 (𝑒 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦)), 𝑣 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ↦ (𝑒 βˆͺ 𝑣)) = (𝑒 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦)), 𝑣 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ↦ (𝑒 βˆͺ 𝑣))
70 vex 3479 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 vsnex 5429 . . . . . . . . . . . . . . . 16 {𝑧} ∈ V
7270, 71unex 7730 . . . . . . . . . . . . . . 15 (𝑦 βˆͺ {𝑧}) ∈ V
7372a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝑦 βˆͺ {𝑧}) ∈ V)
74 simplr 768 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ 𝐹:𝐴⟢Comp)
75 cmptop 22891 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ Comp β†’ π‘₯ ∈ Top)
7675ssriv 3986 . . . . . . . . . . . . . . . 16 Comp βŠ† Top
77 fss 6732 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴⟢Comp ∧ Comp βŠ† Top) β†’ 𝐹:𝐴⟢Top)
7874, 76, 77sylancl 587 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ 𝐹:𝐴⟢Top)
79 simprr 772 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)
8078, 79fssresd 6756 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝐹 β†Ύ (𝑦 βˆͺ {𝑧})):(𝑦 βˆͺ {𝑧})⟢Top)
81 eqidd 2734 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝑦 βˆͺ {𝑧}) = (𝑦 βˆͺ {𝑧}))
82 simprl 770 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ Β¬ 𝑧 ∈ 𝑦)
83 disjsn 4715 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = βˆ… ↔ Β¬ 𝑧 ∈ 𝑦)
8482, 83sylibr 233 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝑦 ∩ {𝑧}) = βˆ…)
8557, 58, 59, 63, 68, 69, 73, 80, 81, 84ptunhmeo 23304 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝑒 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦)), 𝑣 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ↦ (𝑒 βˆͺ 𝑣)) ∈ (((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧})))Homeo(∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧})))))
86 hmphi 23273 . . . . . . . . . . . . 13 ((𝑒 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ 𝑦)), 𝑣 ∈ βˆͺ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ↦ (𝑒 βˆͺ 𝑣)) ∈ (((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧})))Homeo(∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧})))) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ≃ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))))
8785, 86syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ≃ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))))
881ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ 𝐹 Fn 𝐴)
8964, 79sstrid 3993 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ {𝑧} βŠ† 𝐴)
90 vex 3479 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
9190snss 4789 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝐴 ↔ {𝑧} βŠ† 𝐴)
9289, 91sylibr 233 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ 𝑧 ∈ 𝐴)
93 fnressn 7153 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) β†’ (𝐹 β†Ύ {𝑧}) = {βŸ¨π‘§, (πΉβ€˜π‘§)⟩})
9488, 92, 93syl2anc 585 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (𝐹 β†Ύ {𝑧}) = {βŸ¨π‘§, (πΉβ€˜π‘§)⟩})
9594fveq2d 6893 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (∏tβ€˜(𝐹 β†Ύ {𝑧})) = (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}))
96 eqid 2733 . . . . . . . . . . . . . . . . 17 (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}) = (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩})
9790a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ 𝑧 ∈ V)
9874, 92ffvelcdmd 7085 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (πΉβ€˜π‘§) ∈ Comp)
9976, 98sselid 3980 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (πΉβ€˜π‘§) ∈ Top)
100 toptopon2 22412 . . . . . . . . . . . . . . . . . 18 ((πΉβ€˜π‘§) ∈ Top ↔ (πΉβ€˜π‘§) ∈ (TopOnβ€˜βˆͺ (πΉβ€˜π‘§)))
10199, 100sylib 217 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (πΉβ€˜π‘§) ∈ (TopOnβ€˜βˆͺ (πΉβ€˜π‘§)))
10296, 97, 101pt1hmeo 23302 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (π‘₯ ∈ βˆͺ (πΉβ€˜π‘§) ↦ {βŸ¨π‘§, π‘₯⟩}) ∈ ((πΉβ€˜π‘§)Homeo(∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩})))
103 hmphi 23273 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ βˆͺ (πΉβ€˜π‘§) ↦ {βŸ¨π‘§, π‘₯⟩}) ∈ ((πΉβ€˜π‘§)Homeo(∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩})) β†’ (πΉβ€˜π‘§) ≃ (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}))
104102, 103syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (πΉβ€˜π‘§) ≃ (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}))
105 cmphmph 23284 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘§) ≃ (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}) β†’ ((πΉβ€˜π‘§) ∈ Comp β†’ (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}) ∈ Comp))
106104, 98, 105sylc 65 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (∏tβ€˜{βŸ¨π‘§, (πΉβ€˜π‘§)⟩}) ∈ Comp)
10795, 106eqeltrd 2834 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ∈ Comp)
108 txcmp 23139 . . . . . . . . . . . . . 14 (((∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp ∧ (∏tβ€˜(𝐹 β†Ύ {𝑧})) ∈ Comp) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ∈ Comp)
109108expcom 415 . . . . . . . . . . . . 13 ((∏tβ€˜(𝐹 β†Ύ {𝑧})) ∈ Comp β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ∈ Comp))
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ∈ Comp))
111 cmphmph 23284 . . . . . . . . . . . 12 (((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ≃ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) β†’ (((∏tβ€˜(𝐹 β†Ύ 𝑦)) Γ—t (∏tβ€˜(𝐹 β†Ύ {𝑧}))) ∈ Comp β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))
11287, 110, 111sylsyld 61 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) ∧ (Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴)) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))
113112expcom 415 . . . . . . . . . 10 ((Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴) β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ ((∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp)))
114113a2d 29 . . . . . . . . 9 ((Β¬ 𝑧 ∈ 𝑦 ∧ (𝑦 βˆͺ {𝑧}) βŠ† 𝐴) β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp) β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp)))
115114ex 414 . . . . . . . 8 (Β¬ 𝑧 ∈ 𝑦 β†’ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp) β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))))
116115a2d 29 . . . . . . 7 (Β¬ 𝑧 ∈ 𝑦 β†’ (((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)) β†’ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))))
11756, 116syl5 34 . . . . . 6 (Β¬ 𝑧 ∈ 𝑦 β†’ ((𝑦 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)) β†’ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))))
118117adantl 483 . . . . 5 ((𝑦 ∈ Fin ∧ Β¬ 𝑧 ∈ 𝑦) β†’ ((𝑦 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝑦)) ∈ Comp)) β†’ ((𝑦 βˆͺ {𝑧}) βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ (𝑦 βˆͺ {𝑧}))) ∈ Comp))))
11914, 20, 26, 32, 52, 118findcard2s 9162 . . . 4 (𝐴 ∈ Fin β†’ (𝐴 βŠ† 𝐴 β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp)))
1206, 119mpi 20 . . 3 (𝐴 ∈ Fin β†’ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp))
121120anabsi5 668 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜(𝐹 β†Ύ 𝐴)) ∈ Comp)
1225, 121eqeltrrd 2835 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟢Comp) β†’ (∏tβ€˜πΉ) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   β†Ύ cres 5678   Fn wfn 6536  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  Xcixp 8888  Fincfn 8936  βˆtcpt 17381  Topctop 22387  TopOnctopon 22404  Compccmp 22882   Γ—t ctx 23056  Homeochmeo 23249   ≃ chmph 23250
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-topgen 17386  df-pt 17387  df-top 22388  df-topon 22405  df-bases 22441  df-cn 22723  df-cnp 22724  df-cmp 22883  df-tx 23058  df-hmeo 23251  df-hmph 23252
This theorem is referenced by:  poimirlem30  36507
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