Theorem nfrn 5861

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 5600	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		nfrn.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfcnv 5787	. . 3 ⊢ Ⅎ𝑥◡𝐴
4	3	nfdm 5860	. 2 ⊢ Ⅎ𝑥dom ◡𝐴
5	1, 4	nfcxfr 2905	1 ⊢ Ⅎ𝑥ran 𝐴

Syntax hints:  Ⅎwnfc 2887  ^◡ccnv 5588  dom cdm 5589  ran crn 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  nfima  5977  nff  6596  nffo  6687  fliftfun  7183  zfrep6  7797  ptbasfi  22732  utopsnneiplem  23399  restmetu  23726  itg2cnlem1  24926  acunirnmpt2  30997  acunirnmpt2f  30998  fnpreimac  31008  fsumiunle  31143  nsgqusf1olem1  31598  nsgqusf1olem3  31600  locfinreflem  31790  prodindf  31991  esumrnmpt2  32036  esumgect  32058  esum2d  32061  esumiun  32062  sigapildsys  32130  ldgenpisyslem1  32131  oms0  32264  breprexplema  32610  bnj1366  32809  exrecfnlem  35550  totbndbnd  35947  refsumcn  42573  disjrnmpt2  42726  disjf1o  42729  disjinfi  42731  choicefi  42740  rnmptbd2lem  42794  infnsuprnmpt  42796  rnmptbdlem  42801  rnmptss2  42803  rnmptssbi  42807  supxrleubrnmpt  42946  suprleubrnmpt  42962  infrnmptle  42963  infxrunb3rnmpt  42968  uzub  42971  supminfrnmpt  42985  infxrgelbrnmpt  42994  infrpgernmpt  43005  supminfxrrnmpt  43011  limsupubuz  43254  liminflelimsuplem  43316  stoweidlem27  43568  stoweidlem29  43570  stoweidlem31  43572  stoweidlem35  43576  stoweidlem59  43600  stoweidlem62  43603  stirlinglem5  43619  fourierdlem31  43679  fourierdlem80  43727  fourierdlem93  43740  sge00  43914  sge0f1o  43920  sge0gerp  43933  sge0pnffigt  43934  sge0lefi  43936  sge0ltfirp  43938  sge0resplit  43944  sge0reuz  43985  iunhoiioolem  44213  smfpimcc  44341  smfsup  44347  smfsupxr  44349  smfinf  44351  smflimsup  44361  nfafv2  44710