Theorem f2ndf 6383

Description: The 2^nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Assertion

Ref	Expression
f2ndf	⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)

Proof of Theorem f2ndf

Step	Hyp	Ref	Expression
1		f2ndres 6315	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2		fssxp 5496	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3		fssres 5506	. . 3 ⊢ (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
4	1, 2, 3	sylancr 414	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
5		resabs1 5037	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd ↾ 𝐹))
6	2, 5	syl 14	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd ↾ 𝐹))
7	6	eqcomd 2235	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
8	7	feq1d 5463	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ 𝐹):𝐹⟶𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵))
9	4, 8	mpbird 167	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)

Syntax hints:   → wi 4   = wceq 1395   ⊆ wss 3197   × cxp 4718   ↾ cres 4722  ⟶wf 5317  2^nd c2nd 6294

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-2nd 6296

This theorem is referenced by:  fo2ndf  6384  f1o2ndf1  6385