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Theorem ixpfi2 8418
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 3982 . . . 4 (𝐴𝐶) ⊆ 𝐶
3 ssfi 8334 . . . 4 ((𝐶 ∈ Fin ∧ (𝐴𝐶) ⊆ 𝐶) → (𝐴𝐶) ∈ Fin)
41, 2, 3sylancl 574 . . 3 (𝜑 → (𝐴𝐶) ∈ Fin)
5 inss1 3981 . . . 4 (𝐴𝐶) ⊆ 𝐴
6 ixpfi2.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
76ralrimiva 3115 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵 ∈ Fin)
8 ssralv 3815 . . . 4 ((𝐴𝐶) ⊆ 𝐴 → (∀𝑥𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin))
95, 7, 8mpsyl 68 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
10 ixpfi 8417 . . 3 (((𝐴𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
114, 9, 10syl2anc 573 . 2 (𝜑X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin)
12 resixp 8095 . . . . 5 (((𝐴𝐶) ⊆ 𝐴𝑓X𝑥𝐴 𝐵) → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
135, 12mpan 670 . . . 4 (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵)
1413a1i 11 . . 3 (𝜑 → (𝑓X𝑥𝐴 𝐵 → (𝑓 ↾ (𝐴𝐶)) ∈ X𝑥 ∈ (𝐴𝐶)𝐵))
15 simprl 754 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓X𝑥𝐴 𝐵)
16 vex 3354 . . . . . . . . . . 11 𝑓 ∈ V
1716elixp 8067 . . . . . . . . . 10 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1815, 17sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1918simprd 483 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
20 simprr 756 . . . . . . . . . 10 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔X𝑥𝐴 𝐵)
21 vex 3354 . . . . . . . . . . 11 𝑔 ∈ V
2221elixp 8067 . . . . . . . . . 10 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2320, 22sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
2423simprd 483 . . . . . . . 8 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
25 r19.26 3212 . . . . . . . . 9 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
26 difss 3888 . . . . . . . . . . 11 (𝐴𝐶) ⊆ 𝐴
27 ssralv 3815 . . . . . . . . . . 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 3751 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ {𝐷}))
31 elsni 4333 . . . . . . . . . . . . . . 15 ((𝑓𝑥) ∈ {𝐷} → (𝑓𝑥) = 𝐷)
3230, 31syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) = 𝐷))
3329sseld 3751 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) ∈ {𝐷}))
34 elsni 4333 . . . . . . . . . . . . . . 15 ((𝑔𝑥) ∈ {𝐷} → (𝑔𝑥) = 𝐷)
3533, 34syl6 35 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴𝐶)) → ((𝑔𝑥) ∈ 𝐵 → (𝑔𝑥) = 𝐷))
3632, 35anim12d 596 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷)))
37 eqtr3 2792 . . . . . . . . . . . . 13 (((𝑓𝑥) = 𝐷 ∧ (𝑔𝑥) = 𝐷) → (𝑓𝑥) = (𝑔𝑥))
3836, 37syl6 35 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → (((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → (𝑓𝑥) = (𝑔𝑥)))
3938ralimdva 3111 . . . . . . . . . . 11 (𝜑 → (∀𝑥 ∈ (𝐴𝐶)((𝑓𝑥) ∈ 𝐵 ∧ (𝑔𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥)))
4039adantr 466 . . . . . . . . . 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 679 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
4443biantrud 521 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ↔ (∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥) ∧ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))))
45 fvres 6346 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑓 ↾ (𝐴𝐶))‘𝑥) = (𝑓𝑥))
46 fvres 6346 . . . . . . . 8 (𝑥 ∈ (𝐴𝐶) → ((𝑔 ↾ (𝐴𝐶))‘𝑥) = (𝑔𝑥))
4745, 46eqeq12d 2786 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) → (((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ (𝑓𝑥) = (𝑔𝑥)))
4847ralbiia 3128 . . . . . 6 (∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑓𝑥) = (𝑔𝑥))
49 inundif 4188 . . . . . . . 8 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5049raleqi 3291 . . . . . . 7 (∀𝑥 ∈ ((𝐴𝐶) ∪ (𝐴𝐶))(𝑓𝑥) = (𝑔𝑥) ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥))
51 ralunb 3945 . . . . . . 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 482 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑓 Fn 𝐴)
55 fnssres 6142 . . . . . . 7 ((𝑓 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5654, 5, 55sylancl 574 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5723simpld 482 . . . . . . 7 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → 𝑔 Fn 𝐴)
58 fnssres 6142 . . . . . . 7 ((𝑔 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
5957, 5, 58sylancl 574 . . . . . 6 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶))
60 eqfnfv 6452 . . . . . 6 (((𝑓 ↾ (𝐴𝐶)) Fn (𝐴𝐶) ∧ (𝑔 ↾ (𝐴𝐶)) Fn (𝐴𝐶)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
6156, 59, 60syl2anc 573 . . . . 5 ((𝜑 ∧ (𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵)) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ ∀𝑥 ∈ (𝐴𝐶)((𝑓 ↾ (𝐴𝐶))‘𝑥) = ((𝑔 ↾ (𝐴𝐶))‘𝑥)))
62 eqfnfv 6452 . . . . . 6 ((𝑓 Fn 𝐴𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥𝐴 (𝑓𝑥) = (𝑔𝑥)))
6354, 57, 62syl2anc 573 . . . . 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 397 . . 3 (𝜑 → ((𝑓X𝑥𝐴 𝐵𝑔X𝑥𝐴 𝐵) → ((𝑓 ↾ (𝐴𝐶)) = (𝑔 ↾ (𝐴𝐶)) ↔ 𝑓 = 𝑔)))
6614, 65dom2lem 8147 . 2 (𝜑 → (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵)
67 f1fi 8407 . 2 ((X𝑥 ∈ (𝐴𝐶)𝐵 ∈ Fin ∧ (𝑓X𝑥𝐴 𝐵 ↦ (𝑓 ↾ (𝐴𝐶))):X𝑥𝐴 𝐵1-1X𝑥 ∈ (𝐴𝐶)𝐵) → X𝑥𝐴 𝐵 ∈ Fin)
6811, 66, 67syl2anc 573 1 (𝜑X𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cun 3721  cin 3722  wss 3723  {csn 4316  cmpt 4863  cres 5251   Fn wfn 6024  1-1wf1 6026  cfv 6029  Xcixp 8060  Fincfn 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111
This theorem is referenced by:  psrbaglefi  19580  eulerpartlemb  30763
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