Theorem nfrn 4911

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 4674	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		nfrn.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfcnv 4845	. . 3 ⊢ Ⅎ𝑥◡𝐴
4	3	nfdm 4910	. 2 ⊢ Ⅎ𝑥dom ◡𝐴
5	1, 4	nfcxfr 2336	1 ⊢ Ⅎ𝑥ran 𝐴

Syntax hints:  Ⅎwnfc 2326  ^◡ccnv 4662  dom cdm 4663  ran crn 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  nfima  5017  nff  5404  nffo  5479  fliftfun  5843  nfseq  10549