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Theorem f2ndf 5789
 Description: The 2nd (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:AB → (2nd𝐹):𝐹B)

Proof of Theorem f2ndf
StepHypRef Expression
1 f2ndres 5729 . . 3 (2nd ↾ (A × B)):(A × B)⟶B
2 fssxp 5001 . . 3 (𝐹:AB𝐹 ⊆ (A × B))
3 fssres 5009 . . 3 (((2nd ↾ (A × B)):(A × B)⟶B 𝐹 ⊆ (A × B)) → ((2nd ↾ (A × B)) ↾ 𝐹):𝐹B)
41, 2, 3sylancr 393 . 2 (𝐹:AB → ((2nd ↾ (A × B)) ↾ 𝐹):𝐹B)
5 resabs1 4583 . . . . 5 (𝐹 ⊆ (A × B) → ((2nd ↾ (A × B)) ↾ 𝐹) = (2nd𝐹))
62, 5syl 14 . . . 4 (𝐹:AB → ((2nd ↾ (A × B)) ↾ 𝐹) = (2nd𝐹))
76eqcomd 2042 . . 3 (𝐹:AB → (2nd𝐹) = ((2nd ↾ (A × B)) ↾ 𝐹))
87feq1d 4977 . 2 (𝐹:AB → ((2nd𝐹):𝐹B ↔ ((2nd ↾ (A × B)) ↾ 𝐹):𝐹B))
94, 8mpbird 156 1 (𝐹:AB → (2nd𝐹):𝐹B)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ⊆ wss 2911   × cxp 4286   ↾ cres 4290  ⟶wf 4841  2nd 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
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