Theorem f2ndf 8050

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 7946	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2		fssxp 6678	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3		fssres 6689	. . 3 ⊢ (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
4	1, 2, 3	sylancr 587	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵)
5	2	resabs1d 5957	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd ↾ 𝐹))
6	5	eqcomd 2737	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
7	6	feq1d 6633	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ 𝐹):𝐹⟶𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹⟶𝐵))
8	4, 7	mpbird 257	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)

Syntax hints:   → wi 4   ⊆ wss 3902   × cxp 5614   ↾ cres 5618  ⟶wf 6477  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483  df-fn 6484  df-f 6485  df-2nd 7922

This theorem is referenced by:  fo2ndf  8051  f1o2ndf1  8052