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Theorem ixpfi2 9388
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 4246 . . . 4 (𝐴𝐶) ⊆ 𝐶
3 ssfi 9212 . . . 4 ((𝐶 ∈ Fin ∧ (𝐴𝐶) ⊆ 𝐶) → (𝐴𝐶) ∈ Fin)
41, 2, 3sylancl 586 . . 3 (𝜑 → (𝐴𝐶) ∈ Fin)
5 inss1 4245 . . . 4 (𝐴𝐶) ⊆ 𝐴
6 ixpfi2.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
76ralrimiva 3144 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵 ∈ Fin)
8 ssralv 4064 . . . 4 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin))
95, 7, 8mpsyl 68 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
10 ixpfi 9387 . . 3 (((𝐴𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
114, 9, 10syl2anc 584 . 2 (𝜑X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
12 resixp 8972 . . . . 5 (((𝐴𝐶) ⊆ 𝐴𝑓X𝑥𝐴 𝐵) → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
135, 12mpan 690 . . . 4 (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
1413a1i 11 . . 3 (𝜑 → (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵))
15 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓X𝑥𝐴 𝐵)
16 vex 3482 . . . . . . . . . . 11 𝑓 ∈ V
1716elixp 8943 . . . . . . . . . 10 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1815, 17sylib 218 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1918simprd 495 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
20 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔X𝑥𝐴 𝐵)
21 vex 3482 . . . . . . . . . . 11 𝑔 ∈ V
2221elixp 8943 . . . . . . . . . 10 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2320, 22sylib 218 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2423simprd 495 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
25 r19.26 3109 . . . . . . . . 9 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
26 difss 4146 . . . . . . . . . . 11 (𝐴𝐶) ⊆ 𝐴
27 ssralv 4064 . . . . . . . . . . 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 3994 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ {𝐷}))
31 elsni 4648 . . . . . . . . . . . . . . 15 ((𝑓𝑥) ∈ {𝐷} → (𝑓𝑥) = 𝐷)
3230, 31syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) = 𝐷))
3329sseld 3994 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) ∈ {𝐷}))
34 elsni 4648 . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . 13 (((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷) → (𝑓𝑥) = (𝑔𝑥))
3836, 37syl6 35 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → (𝑓𝑥) = (𝑔𝑥)))
3938ralimdva 3165 . . . . . . . . . . 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, 41biimtrrid 243 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4319, 24, 42mp2and 699 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
4443biantrud 531 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))))
45 fvres 6926 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑓 ↾ (𝐴𝐶))‘𝑥) = (𝑓𝑥))
46 fvres 6926 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑔 ↾ (𝐴𝐶))‘𝑥) = (𝑔𝑥))
4745, 46eqeq12d 2751 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) → (((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ (𝑓𝑥) = (𝑔𝑥)))
4847ralbiia 3089 . . . . . 6 (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
49 inundif 4485 . . . . . . . 8 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5049raleqi 3322 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥))
51 ralunb 4207 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5250, 51bitr3i 277 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
5344, 48, 523bitr4g 314 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
5418simpld 494 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓 Fn 𝐴)
55 fnssres 6692 . . . . . . 7 ((𝑓 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5654, 5, 55sylancl 586 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5723simpld 494 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔 Fn 𝐴)
58 fnssres 6692 . . . . . . 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 7051 . . . . . 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 7051 . . . . . 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 311 . . . 4 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔))
6564ex 412 . . 3 (𝜑 → ((𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔)))
6614, 65dom2lem 9031 . 2 (𝜑 → (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵)
67 f1fi 9350 . 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cdif 3960  cun 3961  cin 3962  wss 3963  {csn 4631  cmpt 5231  cres 5691   Fn wfn 6558  1-1wf1 6560  cfv 6563  Xcixp 8936  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  psrbaglefi  21964  eulerpartlemb  34350
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