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Theorem cnvimamptfin 9393
Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 9411, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
Hypothesis
Ref Expression
cnvimamptfin.n (𝜑𝑁 ∈ Fin)
Assertion
Ref Expression
cnvimamptfin (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
Distinct variable group:   𝑁,𝑝
Allowed substitution hints:   𝜑(𝑝)   𝑋(𝑝)   𝑌(𝑝)

Proof of Theorem cnvimamptfin
StepHypRef Expression
1 cnvimamptfin.n . 2 (𝜑𝑁 ∈ Fin)
2 cnvimass 6100 . . 3 ((𝑝𝑁𝑋) “ 𝑌) ⊆ dom (𝑝𝑁𝑋)
3 eqid 2737 . . . 4 (𝑝𝑁𝑋) = (𝑝𝑁𝑋)
43dmmptss 6261 . . 3 dom (𝑝𝑁𝑋) ⊆ 𝑁
52, 4sstri 3993 . 2 ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁
6 ssfi 9213 . 2 ((𝑁 ∈ Fin ∧ ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁) → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
71, 5, 6sylancl 586 1 (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  cmpt 5225  ccnv 5684  dom cdm 5685  cima 5688  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-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-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-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-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by:  elrgspnsubrunlem2  33252
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