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Theorem f2ndf 8044
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 7938 . . 3 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2 fssxp 6693 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
3 fssres 6705 . . 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 5966 . . . 4 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd𝐹))
65eqcomd 2743 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
76feq1d 6650 . 2 (𝐹:𝐴𝐵 → ((2nd𝐹):𝐹𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵))
84, 7mpbird 256 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3908   × cxp 5629  cres 5633  wf 6489  2nd c2nd 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6495  df-fn 6496  df-f 6497  df-2nd 7914
This theorem is referenced by:  fo2ndf  8045  f1o2ndf1  8046
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