Theorem f2ndf 5789

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 5729	. . 3 ⊢ (2nd ↾ (A × B)):(A × B)⟶B
2		fssxp 5001	. . 3 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B))
3		fssres 5009	. . 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 4583	. . . . 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 2042	. . 3 ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹) = ((2nd ↾ (A × B)) ↾ 𝐹))
8	7	feq1d 4977	. 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 1242   ⊆ wss 2911   × cxp 4286   ↾ cres 4290  ⟶wf 4841  2^nd c2nd 5708

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-2nd 5710

This theorem is referenced by:  fo2ndf  5790  f1o2ndf1  5791