Theorem imaeq1 5026

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq1	⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))

Proof of Theorem imaeq1

Step	Hyp	Ref	Expression
1		reseq1 4962	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
2	1	rneqd 4916	. 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶))
3		df-ima 4696	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
4		df-ima 4696	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
5	2, 3, 4	3eqtr4g 2264	1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))

Syntax hints:   → wi 4   = wceq 1373  ran crn 4684   ↾ cres 4685   “ cima 4686

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-cnv 4691  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696

This theorem is referenced by:  imaeq1i  5028  imaeq1d  5030  eceq2  6670  iscnp  14746  elply2  15282