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Theorem fnrndomg 9318
Description: The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
Assertion
Ref Expression
fnrndomg (𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))

Proof of Theorem fnrndomg
StepHypRef Expression
1 dffn4 6088 . 2 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
2 fodomg 9305 . 2 (𝐴𝐵 → (𝐹:𝐴onto→ran 𝐹 → ran 𝐹𝐴))
31, 2syl5bi 232 1 (𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987   class class class wbr 4623  ran crn 5085   Fn wfn 5852  ontowfo 5855  cdom 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-ac2 9245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-card 8725  df-acn 8728  df-ac 8899
This theorem is referenced by:  fnct  9319  unirnfdomd  9349  konigthlem  9350  abrexdomjm  29233  ffsrn  29388  abrexdom  33196  indexdom  33200  subsaliuncl  39913  omeiunle  40068  smflimlem6  40321
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