Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvimamptfin Structured version   Visualization version   GIF version

Theorem cnvimamptfin 8150
 Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8166, 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 5404 . . 3 ((𝑝𝑁𝑋) “ 𝑌) ⊆ dom (𝑝𝑁𝑋)
3 eqid 2610 . . . 4 (𝑝𝑁𝑋) = (𝑝𝑁𝑋)
43dmmptss 5548 . . 3 dom (𝑝𝑁𝑋) ⊆ 𝑁
52, 4sstri 3577 . 2 ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁
6 ssfi 8065 . 2 ((𝑁 ∈ Fin ∧ ((𝑝𝑁𝑋) “ 𝑌) ⊆ 𝑁) → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
71, 5, 6sylancl 693 1 (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-om 6958  df-er 7629  df-en 7842  df-fin 7845 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator