Theorem fv0p1e1 8859

Description: Function value at N + 1 with N replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)

Assertion

Ref	Expression
fv0p1e1	|- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Proof of Theorem fv0p1e1

Step	Hyp	Ref	Expression
1		oveq1 5789	. . 3 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
2		0p1e1 8858	. . 3 |- ( 0 + 1 ) = 1
3	1, 2	eqtrdi 2189	. 2 |- ( N = 0 -> ( N + 1 ) = 1 )
4	3	fveq2d 5433	1 |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Syntax hints:   -> wi 4   = wceq 1332   ` cfv 5131  (class class class)co 5782   0cc0 7644   1c1 7645   + caddc 7647

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-mulcl 7742  ax-addcom 7744  ax-i2m1 7749  ax-0id 7752

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  mertenslem2  11337