Theorem fv0p1e1 8693

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 5713	. . 3 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1))
2		0p1e1 8692	. . 3 ⊢ (0 + 1) = 1
3	1, 2	syl6eq 2148	. 2 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1)
4	3	fveq2d 5357	1 ⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))

Syntax hints:   → wi 4   = wceq 1299  ‘cfv 5059  (class class class)co 5706  0cc0 7500  1c1 7501   + caddc 7503

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-1cn 7588  ax-icn 7590  ax-addcl 7591  ax-mulcl 7593  ax-addcom 7595  ax-i2m1 7600  ax-0id 7603

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709

This theorem is referenced by:  mertenslem2  11144