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Theorem f2ndf 5874
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 5814 . . 3 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
2 fssxp 5085 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
3 fssres 5093 . . 3 (((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵𝐹 ⊆ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
41, 2, 3sylancr 399 . 2 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵)
5 resabs1 4667 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd𝐹))
62, 5syl 14 . . . 4 (𝐹:𝐴𝐵 → ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹) = (2nd𝐹))
76eqcomd 2061 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹) = ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹))
87feq1d 5061 . 2 (𝐹:𝐴𝐵 → ((2nd𝐹):𝐹𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)) ↾ 𝐹):𝐹𝐵))
94, 8mpbird 160 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2944   × cxp 4370  cres 4374  wf 4925  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937  df-2nd 5795
This theorem is referenced by:  fo2ndf  5875  f1o2ndf1  5876
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