Theorem f2ndf 8062

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 7958	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2		fssxp 6689	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3		fssres 6700	. . 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 5967	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd ↾ 𝐹))
6	5	eqcomd 2742	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
7	6	feq1d 6644	. 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 3901   × cxp 5622   ↾ cres 5626  ⟶wf 6488  2^nd c2nd 7932

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496  df-2nd 7934

This theorem is referenced by:  fo2ndf  8063  f1o2ndf1  8064