Theorem feq123 5505

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 1024	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐹 = 𝐺)
2		simp2 1025	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐶)
3		simp3 1026	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
4	1, 2, 3	feq123d 5504	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ⟶wf 5353

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361

This theorem is referenced by: (None)