Theorem feq123 6681

Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)

Assertion

Ref	Expression
feq123	⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))

Proof of Theorem feq123

Step	Hyp	Ref	Expression
1		simp1 1149	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐹 = 𝐺)
2		simp2 1150	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐶)
3		simp3 1151	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
4	1, 2, 3	feq123d 6680	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1098   = wceq 1560  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-fun 6523  df-fn 6524  df-f 6525

This theorem is referenced by:  feq12i  6684  hashfxnn0  14350  mbfresfi  38162