Theorem f1ssr 6343

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 6338	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2	1	adantr 474	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹 Fn 𝐴)
3		simpr 479	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐶)
4		df-f 6126	. . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	sylanbrc 580	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)
6		df-f1 6127	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
7	6	simprbi 492	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
8	7	adantr 474	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → Fun ◡𝐹)
9		df-f1 6127	. 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹))
10	5, 8, 9	sylanbrc 580	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)

Syntax hints:   → wi 4   ∧ wa 386   ⊆ wss 3797  ^◡ccnv 5340  ran crn 5342  Fun wfun 6116   Fn wfn 6117  ⟶wf 6118  –1-1→wf1 6119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 199  df-an 387  df-f 6126  df-f1 6127

This theorem is referenced by:  f1resf1  6345  domdifsn  8311  marypha1  8608  m2cpmf1  20917  ausgrusgri  26466  uspgrupgrushgr  26475  usgrumgruspgr  26478  usgruspgrb  26479  usgrres  26604  usgrres1  26611