Theorem nfrn 5818

Description: Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfrn.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrn	⊢ Ⅎ𝑥ran 𝐴

Proof of Theorem nfrn

Step	Hyp	Ref	Expression
1		df-rn 5560	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		nfrn.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfcnv 5743	. . 3 ⊢ Ⅎ𝑥◡𝐴
4	3	nfdm 5817	. 2 ⊢ Ⅎ𝑥dom ◡𝐴
5	1, 4	nfcxfr 2975	1 ⊢ Ⅎ𝑥ran 𝐴

Syntax hints:  Ⅎwnfc 2961  ^◡ccnv 5548  dom cdm 5549  ran crn 5550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560

This theorem is referenced by:  nfima  5931  nff  6504  nffo  6583  fliftfun  7059  zfrep6  7650  ptbasfi  22183  utopsnneiplem  22850  restmetu  23174  itg2cnlem1  24356  acunirnmpt2  30399  acunirnmpt2f  30400  fnpreimac  30410  fsumiunle  30540  locfinreflem  31099  prodindf  31277  esumrnmpt2  31322  esumgect  31344  esum2d  31347  esumiun  31348  sigapildsys  31416  ldgenpisyslem1  31417  oms0  31550  breprexplema  31896  bnj1366  32096  exrecfnlem  34654  totbndbnd  35061  refsumcn  41280  suprnmpt  41423  disjrnmpt2  41442  disjf1o  41445  disjinfi  41447  choicefi  41456  rnmptbd2lem  41513  infnsuprnmpt  41515  rnmptbdlem  41520  rnmptss2  41522  rnmptssbi  41527  supxrleubrnmpt  41672  suprleubrnmpt  41689  infrnmptle  41690  infxrunb3rnmpt  41695  uzub  41698  supminfrnmpt  41712  infxrgelbrnmpt  41723  infrpgernmpt  41734  supminfxrrnmpt  41740  limsupubuz  41987  liminflelimsuplem  42049  stoweidlem27  42306  stoweidlem29  42308  stoweidlem31  42310  stoweidlem35  42314  stoweidlem59  42338  stoweidlem62  42341  stirlinglem5  42357  fourierdlem31  42417  fourierdlem53  42438  fourierdlem80  42465  fourierdlem93  42478  sge00  42652  sge0f1o  42658  sge0gerp  42671  sge0pnffigt  42672  sge0lefi  42674  sge0ltfirp  42676  sge0resplit  42682  sge0reuz  42723  iunhoiioolem  42951  smfpimcc  43076  smfsup  43082  smfsupxr  43084  smfinf  43086  smflimsup  43096  nfafv2  43411