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Theorem ixpfi2 9333
Description: A Cartesian product of finite sets such that all but finitely many are singletons is finite. (Note that 𝐵(𝑥) and 𝐷(𝑥) are both possibly dependent on 𝑥.) (Contributed by Mario Carneiro, 25-Jan-2015.)
Hypotheses
Ref Expression
ixpfi2.1 (𝜑𝐶 ∈ Fin)
ixpfi2.2 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
ixpfi2.3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵 ⊆ {𝐷})
Assertion
Ref Expression
ixpfi2 (𝜑X𝑥𝐴 𝐵 ∈ Fin)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem ixpfi2
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ixpfi2.1 . . . 4 (𝜑𝐶 ∈ Fin)
2 inss2 4225 . . . 4 (𝐴𝐶) ⊆ 𝐶
3 ssfi 9156 . . . 4 ((𝐶 ∈ Fin ∧ (𝐴𝐶) ⊆ 𝐶) → (𝐴𝐶) ∈ Fin)
41, 2, 3sylancl 586 . . 3 (𝜑 → (𝐴𝐶) ∈ Fin)
5 inss1 4224 . . . 4 (𝐴𝐶) ⊆ 𝐴
6 ixpfi2.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
76ralrimiva 3145 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵 ∈ Fin)
8 ssralv 4046 . . . 4 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin))
95, 7, 8mpsyl 68 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
10 ixpfi 9332 . . 3 (((𝐴𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
114, 9, 10syl2anc 584 . 2 (𝜑X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
12 resixp 8910 . . . . 5 (((𝐴𝐶) ⊆ 𝐴𝑓X𝑥𝐴 𝐵) → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
135, 12mpan 688 . . . 4 (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
1413a1i 11 . . 3 (𝜑 → (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵))
15 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓X𝑥𝐴 𝐵)
16 vex 3477 . . . . . . . . . . 11 𝑓 ∈ V
1716elixp 8881 . . . . . . . . . 10 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1815, 17sylib 217 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1918simprd 496 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
20 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔X𝑥𝐴 𝐵)
21 vex 3477 . . . . . . . . . . 11 𝑔 ∈ V
2221elixp 8881 . . . . . . . . . 10 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2320, 22sylib 217 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2423simprd 496 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
25 r19.26 3110 . . . . . . . . 9 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
26 difss 4127 . . . . . . . . . . 11 (𝐴𝐶) ⊆ 𝐴
27 ssralv 4046 . . . . . . . . . . 11 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵)))
2826, 27ax-mp 5 . . . . . . . . . 10 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵))
29 ixpfi2.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵 ⊆ {𝐷})
3029sseld 3977 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ {𝐷}))
31 elsni 4639 . . . . . . . . . . . . . . 15 ((𝑓𝑥) ∈ {𝐷} → (𝑓𝑥) = 𝐷)
3230, 31syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) = 𝐷))
3329sseld 3977 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) ∈ {𝐷}))
34 elsni 4639 . . . . . . . . . . . . . . 15 ((𝑔𝑥) ∈ {𝐷} → (𝑔𝑥) = 𝐷)
3533, 34syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) = 𝐷))
3632, 35anim12d 609 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷)))
37 eqtr3 2757 . . . . . . . . . . . . 13 (((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷) → (𝑓𝑥) = (𝑔𝑥))
3836, 37syl6 35 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → (𝑓𝑥) = (𝑔𝑥)))
3938ralimdva 3166 . . . . . . . . . . 11 (𝜑 → (∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4039adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4128, 40syl5 34 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4225, 41biimtrrid 242 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4319, 24, 42mp2and 697 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
4443biantrud 532 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))))
45 fvres 6897 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑓 ↾ (𝐴𝐶))‘𝑥) = (𝑓𝑥))
46 fvres 6897 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑔 ↾ (𝐴𝐶))‘𝑥) = (𝑔𝑥))
4745, 46eqeq12d 2747 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) → (((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ (𝑓𝑥) = (𝑔𝑥)))
4847ralbiia 3090 . . . . . 6 (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
49 inundif 4474 . . . . . . . 8 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5049raleqi 3322 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥))
51 ralunb 4187 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5250, 51bitr3i 276 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5344, 48, 523bitr4g 313 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
5418simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓 Fn 𝐴)
55 fnssres 6660 . . . . . . 7 ((𝑓 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5654, 5, 55sylancl 586 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5723simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔 Fn 𝐴)
58 fnssres 6660 . . . . . . 7 ((𝑔 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5957, 5, 58sylancl 586 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
60 eqfnfv 7018 . . . . . 6 (((𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶) ∧ (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
6156, 59, 60syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
62 eqfnfv 7018 . . . . . 6 ((𝑓 Fn 𝐴𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
6354, 57, 62syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 = 𝑔 ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
6453, 61, 633bitr4d 310 . . . 4 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔))
6564ex 413 . . 3 (𝜑 → ((𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔)))
6614, 65dom2lem 8971 . 2 (𝜑 → (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵)
67 f1fi 9322 . 2 ((X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin ∧ (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵) → X𝑥𝐴 𝐵 ∈ Fin)
6811, 66, 67syl2anc 584 1 (𝜑X𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  cdif 3941  cun 3942  cin 3943  wss 3944  {csn 4622  cmpt 5224  cres 5671   Fn wfn 6527  1-1wf1 6529  cfv 6532  Xcixp 8874  Fincfn 8922
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-1o 8448  df-er 8686  df-map 8805  df-pm 8806  df-ixp 8875  df-en 8923  df-dom 8924  df-fin 8926
This theorem is referenced by:  psrbaglefi  21416  psrbaglefiOLD  21417  eulerpartlemb  33198
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