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Theorem 2ndcof 6062
 Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
2ndcof (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)

Proof of Theorem 2ndcof
StepHypRef Expression
1 fo2nd 6056 . . . 4 2nd :V–onto→V
2 fofn 5347 . . . 4 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . 3 2nd Fn V
4 ffn 5272 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5274 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 121 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5297 . . 3 ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd𝐹) Fn 𝐴)
83, 6, 7sylancr 410 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹) Fn 𝐴)
9 rnco 5045 . . 3 ran (2nd𝐹) = ran (2nd ↾ ran 𝐹)
10 frn 5281 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4846 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12 rnss 4769 . . . . 5 ((2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
1310, 11, 123syl 17 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
14 f2ndres 6058 . . . . 5 (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15 frn 5281 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶)
1614, 15ax-mp 5 . . . 4 ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶
1713, 16sstrdi 3109 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
189, 17eqsstrid 3143 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd𝐹) ⊆ 𝐶)
19 df-f 5127 . 2 ((2nd𝐹):𝐴𝐶 ↔ ((2nd𝐹) Fn 𝐴 ∧ ran (2nd𝐹) ⊆ 𝐶))
208, 18, 19sylanbrc 413 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4  Vcvv 2686   ⊆ wss 3071   × cxp 4537  ran crn 4540   ↾ cres 4541   ∘ ccom 4543   Fn wfn 5118  ⟶wf 5119  –onto→wfo 5121  2nd c2nd 6037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-2nd 6039 This theorem is referenced by: (None)
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