Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f2ndf Structured version   Visualization version   GIF version

Theorem f2ndf 7439
 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 7346 . . 3 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2 fssxp 6209 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
3 fssres 6219 . . 3 (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
41, 2, 3sylancr 698 . 2 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
52resabs1d 5574 . . . 4 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd𝐹))
65eqcomd 2754 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
76feq1d 6179 . 2 (𝐹:𝐴𝐵 → ((2nd𝐹):𝐹𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵))
84, 7mpbird 247 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3703   × cxp 5252   ↾ cres 5256  ⟶wf 6033  2nd c2nd 7320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-2nd 7322 This theorem is referenced by:  fo2ndf  7440  f1o2ndf1  7441
 Copyright terms: Public domain W3C validator