Theorem nfrn 4828

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 4594	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		nfrn.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfcnv 4762	. . 3 ⊢ Ⅎ𝑥◡𝐴
4	3	nfdm 4827	. 2 ⊢ Ⅎ𝑥dom ◡𝐴
5	1, 4	nfcxfr 2296	1 ⊢ Ⅎ𝑥ran 𝐴

Syntax hints:  Ⅎwnfc 2286  ^◡ccnv 4582  dom cdm 4583  ran crn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4591  df-dm 4593  df-rn 4594

This theorem is referenced by:  nfima  4933  nff  5313  nffo  5388  fliftfun  5741  nfseq  10336