Theorem fv0p1e1 8635

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 5697	. . 3 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
2		0p1e1 8634	. . 3 |- ( 0 + 1 ) = 1
3	1, 2	syl6eq 2143	. 2 |- ( N = 0 -> ( N + 1 ) = 1 )
4	3	fveq2d 5344	1 |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )

Syntax hints:   -> wi 4   = wceq 1296   ` cfv 5049  (class class class)co 5690   0cc0 7447   1c1 7448   + caddc 7450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-1cn 7535  ax-icn 7537  ax-addcl 7538  ax-mulcl 7540  ax-addcom 7542  ax-i2m1 7547  ax-0id 7550

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-iota 5014  df-fv 5057  df-ov 5693

This theorem is referenced by:  mertenslem2  11094