Theorem nfrn 5932

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 5665	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		nfrn.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfcnv 5858	. . 3 ⊢ Ⅎ𝑥◡𝐴
4	3	nfdm 5931	. 2 ⊢ Ⅎ𝑥dom ◡𝐴
5	1, 4	nfcxfr 2896	1 ⊢ Ⅎ𝑥ran 𝐴

Syntax hints:  Ⅎwnfc 2883  ^◡ccnv 5653  dom cdm 5654  ran crn 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by:  nfima  6055  nff  6702  nffo  6789  fliftfun  7305  zfrep6  7953  ptbasfi  23519  utopsnneiplem  24186  restmetu  24509  itg2cnlem1  25714  acunirnmpt2  32638  acunirnmpt2f  32639  fnpreimac  32649  fsumiunle  32808  prodindf  32840  nsgqusf1olem1  33428  nsgqusf1olem3  33430  locfinreflem  33871  esumrnmpt2  34099  esumgect  34121  esum2d  34124  esumiun  34125  sigapildsys  34193  ldgenpisyslem1  34194  oms0  34329  breprexplema  34662  bnj1366  34860  exrecfnlem  37397  totbndbnd  37813  modelaxreplem3  45005  refsumcn  45054  disjrnmpt2  45212  disjf1o  45215  disjinfi  45216  choicefi  45224  rnmptbd2lem  45272  infnsuprnmpt  45274  rnmptbdlem  45279  rnmptss2  45281  rnmptssbi  45284  supxrleubrnmpt  45433  suprleubrnmpt  45449  infrnmptle  45450  infxrunb3rnmpt  45455  uzub  45458  supminfrnmpt  45472  infxrgelbrnmpt  45481  infrpgernmpt  45492  supminfxrrnmpt  45498  limsupubuz  45742  liminflelimsuplem  45804  stoweidlem27  46056  stoweidlem29  46058  stoweidlem31  46060  stoweidlem35  46064  stoweidlem59  46088  stoweidlem62  46091  stirlinglem5  46107  fourierdlem31  46167  fourierdlem80  46215  fourierdlem93  46228  sge00  46405  sge0f1o  46411  sge0gerp  46424  sge0pnffigt  46425  sge0lefi  46427  sge0ltfirp  46429  sge0resplit  46435  sge0reuz  46476  iunhoiioolem  46704  smfpimcc  46837  smfsup  46843  smfsupxr  46845  smfinf  46847  smflimsup  46857  nfafv2  47247