Theorem fv0p1e1 8838

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 5781	. . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2		0p1e1 8837	. . 3 ⊢ (0 + 1) = 1
3	1, 2	syl6eq 2188	. 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1)
4	3	fveq2d 5425	1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))

Syntax hints:   → wi 4   = wceq 1331  ‘cfv 5123  (class class class)co 5774  0cc0 7623  1c1 7624   + caddc 7626

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-1cn 7716  ax-icn 7718  ax-addcl 7719  ax-mulcl 7721  ax-addcom 7723  ax-i2m1 7728  ax-0id 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  mertenslem2  11308