Theorem nfrn 5004

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	|- F/_ x A

Assertion

Ref	Expression
nfrn	|- F/_ x ran A

Proof of Theorem nfrn

Step	Hyp	Ref	Expression
1		df-rn 4762	. 2 |- ran A = dom `' A
2		nfrn.1	. . . 4 |- F/_ x A
3	2	nfcnv 4936	. . 3 |- F/_ x `' A
4	3	nfdm 5003	. 2 |- F/_ x dom `' A
5	1, 4	nfcxfr 2383	1 |- F/_ x ran A

Syntax hints:   F/_wnfc 2373   `'ccnv 4750   dom cdm 4751   ran crn 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by:  nfima  5111  nff  5507  nffo  5591  fliftfun  5971  nfseq  10823