Theorem f1ssr 5410

Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Assertion

Ref	Expression
f1ssr	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)

Proof of Theorem f1ssr

Step	Hyp	Ref	Expression
1		f1fn 5405	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2	1	adantr 274	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹 Fn 𝐴)
3		simpr 109	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐶)
4		df-f 5202	. . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	sylanbrc 415	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)
6		df-f1 5203	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
7	6	simprbi 273	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
8	7	adantr 274	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → Fun ◡𝐹)
9		df-f1 5203	. 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹))
10	5, 8, 9	sylanbrc 415	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ⊆ wss 3121  ^◡ccnv 4610  ran crn 4612  Fun wfun 5192   Fn wfn 5193  ⟶wf 5194  –1-1→wf1 5195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-f 5202  df-f1 5203

This theorem is referenced by:  f1ff1  5411  difinfsn  7077