Theorem f1ssr 6778

Description: A function that is one-to-one is also one-to-one on some superset of its range. (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 6771	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2	1	adantr 486	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹 Fn 𝐴)
3		simpr 490	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐶)
4		df-f 6535	. . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	sylanbrc 595	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)
6		df-f1 6536	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
7	6	simprbi 503	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
8	7	adantr 486	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → Fun ◡𝐹)
9		df-f1 6536	. 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹))
10	5, 8, 9	sylanbrc 595	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3899  ^◡ccnv 5650  ran crn 5652  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-f 6535  df-f1 6536

This theorem is used by:  f1resf1  6780  domdifsn  9063  marypha1  9410  m2cpmf1  23041  ausgrusgri  29731  uspgrupgrushgr  29742  usgrumgruspgr  29745  usgruspgrb  29746  usgrres  29871  usgrres1  29878  lindflbs  33916  dimkerim  34241  fineqvinfep  35766  cantnfub2  44282