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

Theorem f2ndf 8144
Description: The 2nd (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:𝐴𝐵 → (2nd𝐹):𝐹𝐵)

Proof of Theorem f2ndf
StepHypRef Expression
1 f2ndres 8038 . . 3 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2 fssxp 6764 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
3 fssres 6775 . . 3 (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
41, 2, 3sylancr 587 . 2 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
52resabs1d 6028 . . . 4 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd𝐹))
65eqcomd 2741 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
76feq1d 6721 . 2 (𝐹:𝐴𝐵 → ((2nd𝐹):𝐹𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵))
84, 7mpbird 257 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3963   × cxp 5687  cres 5691  wf 6559  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-2nd 8014
This theorem is referenced by:  fo2ndf  8145  f1o2ndf1  8146
  Copyright terms: Public domain W3C validator