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Theorem ixpfi2 9390
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 4238 . . . 4 (𝐴𝐶) ⊆ 𝐶
3 ssfi 9213 . . . 4 ((𝐶 ∈ Fin ∧ (𝐴𝐶) ⊆ 𝐶) → (𝐴𝐶) ∈ Fin)
41, 2, 3sylancl 586 . . 3 (𝜑 → (𝐴𝐶) ∈ Fin)
5 inss1 4237 . . . 4 (𝐴𝐶) ⊆ 𝐴
6 ixpfi2.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
76ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵 ∈ Fin)
8 ssralv 4052 . . . 4 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin))
95, 7, 8mpsyl 68 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
10 ixpfi 9389 . . 3 (((𝐴𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
114, 9, 10syl2anc 584 . 2 (𝜑X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
12 resixp 8973 . . . . 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 3484 . . . . . . . . . . 11 𝑓 ∈ V
1716elixp 8944 . . . . . . . . . 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 3484 . . . . . . . . . . 11 𝑔 ∈ V
2221elixp 8944 . . . . . . . . . 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 3111 . . . . . . . . 9 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
26 difss 4136 . . . . . . . . . . 11 (𝐴𝐶) ⊆ 𝐴
27 ssralv 4052 . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ {𝐷}))
31 elsni 4643 . . . . . . . . . . . . . . 15 ((𝑓𝑥) ∈ {𝐷} → (𝑓𝑥) = 𝐷)
3230, 31syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) = 𝐷))
3329sseld 3982 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) ∈ {𝐷}))
34 elsni 4643 . . . . . . . . . . . . . . 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 2763 . . . . . . . . . . . . 13 (((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷) → (𝑓𝑥) = (𝑔𝑥))
3836, 37syl6 35 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → (𝑓𝑥) = (𝑔𝑥)))
3938ralimdva 3167 . . . . . . . . . . 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 6925 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑓 ↾ (𝐴𝐶))‘𝑥) = (𝑓𝑥))
46 fvres 6925 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑔 ↾ (𝐴𝐶))‘𝑥) = (𝑔𝑥))
4745, 46eqeq12d 2753 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) → (((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ (𝑓𝑥) = (𝑔𝑥)))
4847ralbiia 3091 . . . . . 6 (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
49 inundif 4479 . . . . . . . 8 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5049raleqi 3324 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥))
51 ralunb 4197 . . . . . . 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 6691 . . . . . . 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 6691 . . . . . . 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 9032 . 2 (𝜑 → (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵)
67 f1fi 9352 . 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 1540  wcel 2108  wral 3061  cdif 3948  cun 3949  cin 3950  wss 3951  {csn 4626  cmpt 5225  cres 5687   Fn wfn 6556  1-1wf1 6558  cfv 6561  Xcixp 8937  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989
This theorem is referenced by:  psrbaglefi  21946  eulerpartlemb  34370
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