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Theorem ixpfi2 8305
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 3867 . . . 4 (𝐴𝐶) ⊆ 𝐶
3 ssfi 8221 . . . 4 ((𝐶 ∈ Fin ∧ (𝐴𝐶) ⊆ 𝐶) → (𝐴𝐶) ∈ Fin)
41, 2, 3sylancl 695 . . 3 (𝜑 → (𝐴𝐶) ∈ Fin)
5 inss1 3866 . . . 4 (𝐴𝐶) ⊆ 𝐴
6 ixpfi2.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
76ralrimiva 2995 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵 ∈ Fin)
8 ssralv 3699 . . . 4 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin))
95, 7, 8mpsyl 68 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
10 ixpfi 8304 . . 3 (((𝐴𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
114, 9, 10syl2anc 694 . 2 (𝜑X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
12 resixp 7985 . . . . 5 (((𝐴𝐶) ⊆ 𝐴𝑓X𝑥𝐴 𝐵) → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
135, 12mpan 706 . . . 4 (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
1413a1i 11 . . 3 (𝜑 → (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵))
15 simprl 809 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓X𝑥𝐴 𝐵)
16 vex 3234 . . . . . . . . . . 11 𝑓 ∈ V
1716elixp 7957 . . . . . . . . . 10 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1815, 17sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1918simprd 478 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
20 simprr 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔X𝑥𝐴 𝐵)
21 vex 3234 . . . . . . . . . . 11 𝑔 ∈ V
2221elixp 7957 . . . . . . . . . 10 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2320, 22sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2423simprd 478 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
25 r19.26 3093 . . . . . . . . 9 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
26 difss 3770 . . . . . . . . . . 11 (𝐴𝐶) ⊆ 𝐴
27 ssralv 3699 . . . . . . . . . . 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 3635 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ {𝐷}))
31 elsni 4227 . . . . . . . . . . . . . . 15 ((𝑓𝑥) ∈ {𝐷} → (𝑓𝑥) = 𝐷)
3230, 31syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) = 𝐷))
3329sseld 3635 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) ∈ {𝐷}))
34 elsni 4227 . . . . . . . . . . . . . . 15 ((𝑔𝑥) ∈ {𝐷} → (𝑔𝑥) = 𝐷)
3533, 34syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) = 𝐷))
3632, 35anim12d 585 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷)))
37 eqtr3 2672 . . . . . . . . . . . . 13 (((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷) → (𝑓𝑥) = (𝑔𝑥))
3836, 37syl6 35 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → (𝑓𝑥) = (𝑔𝑥)))
3938ralimdva 2991 . . . . . . . . . . 11 (𝜑 → (∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4039adantr 480 . . . . . . . . . 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, 41syl5bir 233 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4319, 24, 42mp2and 715 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
4443biantrud 527 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))))
45 fvres 6245 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑓 ↾ (𝐴𝐶))‘𝑥) = (𝑓𝑥))
46 fvres 6245 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑔 ↾ (𝐴𝐶))‘𝑥) = (𝑔𝑥))
4745, 46eqeq12d 2666 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) → (((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ (𝑓𝑥) = (𝑔𝑥)))
4847ralbiia 3008 . . . . . 6 (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
49 inundif 4079 . . . . . . . 8 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5049raleqi 3172 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥))
51 ralunb 3827 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5250, 51bitr3i 266 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5344, 48, 523bitr4g 303 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
5418simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓 Fn 𝐴)
55 fnssres 6042 . . . . . . 7 ((𝑓 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5654, 5, 55sylancl 695 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5723simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔 Fn 𝐴)
58 fnssres 6042 . . . . . . 7 ((𝑔 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5957, 5, 58sylancl 695 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
60 eqfnfv 6351 . . . . . 6 (((𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶) ∧ (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
6156, 59, 60syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
62 eqfnfv 6351 . . . . . 6 ((𝑓 Fn 𝐴𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
6354, 57, 62syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 = 𝑔 ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
6453, 61, 633bitr4d 300 . . . 4 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔))
6564ex 449 . . 3 (𝜑 → ((𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔)))
6614, 65dom2lem 8037 . 2 (𝜑 → (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵)
67 f1fi 8294 . 2 ((X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin ∧ (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵) → X𝑥𝐴 𝐵 ∈ Fin)
6811, 66, 67syl2anc 694 1 (𝜑X𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  cun 3605  cin 3606  wss 3607  {csn 4210  cmpt 4762  cres 5145   Fn wfn 5921  1-1wf1 5923  cfv 5926  Xcixp 7950  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001
This theorem is referenced by:  psrbaglefi  19420  eulerpartlemb  30558
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