Theorem f2ndf 6229

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 6163	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2		fssxp 5385	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3		fssres 5393	. . 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 4938	. . . . 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 2183	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
8	7	feq1d 5354	. 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 1353   ⊆ wss 3131   × cxp 4626   ↾ cres 4630  ⟶wf 5214  2^nd c2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-2nd 6144

This theorem is referenced by:  fo2ndf  6230  f1o2ndf1  6231