Theorem f2ndf 5847

Description: The 2^nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 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 5787	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2		fssxp 5058	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3		fssres 5066	. . 3 ⊢ (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
4	1, 2, 3	sylancr 393	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
5		resabs1 4640	. . . . 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 2045	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
8	7	feq1d 5034	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ 𝐹):𝐹⟶𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵))
9	4, 8	mpbird 156	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917   × cxp 4343   ↾ cres 4347  ⟶wf 4898  2^nd c2nd 5766

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-2nd 5768

This theorem is referenced by:  fo2ndf  5848  f1o2ndf1  5849