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Theorem rnmptfi 41295
Description: The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
rnmptfi.a 𝐴 = (𝑥𝐵𝐶)
Assertion
Ref Expression
rnmptfi (𝐵 ∈ Fin → ran 𝐴 ∈ Fin)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem rnmptfi
StepHypRef Expression
1 rnmptfi.a . . 3 𝐴 = (𝑥𝐵𝐶)
2 mptfi 8812 . . 3 (𝐵 ∈ Fin → (𝑥𝐵𝐶) ∈ Fin)
31, 2eqeltrid 2922 . 2 (𝐵 ∈ Fin → 𝐴 ∈ Fin)
4 rnfi 8796 . 2 (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
53, 4syl 17 1 (𝐵 ∈ Fin → ran 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cmpt 5143  ran crn 5555  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-fin 8502
This theorem is referenced by:  fisupclrnmpt  41539  stoweidlem35  42189  fourierdlem50  42310  fourierdlem70  42330  fourierdlem71  42331  fourierdlem76  42336  fourierdlem80  42340  fourierdlem103  42363  fourierdlem104  42364  ioorrnopnlem  42458  hoidmvlelem2  42747  iunhoiioolem  42826  vonioolem1  42831
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