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Theorem cnvimamptfin 9367
Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 9384, 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 6079 . . 3 ((𝑝𝑁𝑋) “ 𝑌) ⊆ dom (𝑝𝑁𝑋)
3 eqid 2727 . . . 4 (𝑝𝑁𝑋) = (𝑝𝑁𝑋)
43dmmptss 6239 . . 3 dom (𝑝𝑁𝑋) ⊆ 𝑁
52, 4sstri 3987 . 2 ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁
6 ssfi 9187 . 2 ((𝑁 ∈ Fin ∧ ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁) → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
71, 5, 6sylancl 585 1 (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3944  cmpt 5225  ccnv 5671  dom cdm 5672  cima 5675  Fincfn 8953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7863  df-1o 8478  df-en 8954  df-fin 8957
This theorem is referenced by: (None)
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