Theorem fss 4997

Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fss	⊢ ((𝐹:A⟶B ∧ B ⊆ 𝐶) → 𝐹:A⟶𝐶)

Proof of Theorem fss

Step	Hyp	Ref	Expression
1		sstr2 2946	. . . . 5 ⊢ (ran 𝐹 ⊆ B → (B ⊆ 𝐶 → ran 𝐹 ⊆ 𝐶))
2	1	com12 27	. . . 4 ⊢ (B ⊆ 𝐶 → (ran 𝐹 ⊆ B → ran 𝐹 ⊆ 𝐶))
3	2	anim2d 320	. . 3 ⊢ (B ⊆ 𝐶 → ((𝐹 Fn A ∧ ran 𝐹 ⊆ B) → (𝐹 Fn A ∧ ran 𝐹 ⊆ 𝐶)))
4		df-f 4849	. . 3 ⊢ (𝐹:A⟶B ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ B))
5		df-f 4849	. . 3 ⊢ (𝐹:A⟶𝐶 ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ 𝐶))
6	3, 4, 5	3imtr4g 194	. 2 ⊢ (B ⊆ 𝐶 → (𝐹:A⟶B → 𝐹:A⟶𝐶))
7	6	impcom 116	1 ⊢ ((𝐹:A⟶B ∧ B ⊆ 𝐶) → 𝐹:A⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849

This theorem is referenced by:  fssd  4998  fconst6g  5028  f1ss  5040  ffoss  5101  fsn2  5280  ofco  5671  tposf2  5824  issmo2  5845  smoiso  5858  ssdomg  6194  1fv  8766