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Theorem 2ndcof 5949
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 5943 . . . 4 2nd :V–onto→V
2 fofn 5248 . . . 4 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 7 . . 3 2nd Fn V
4 ffn 5174 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5176 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 121 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5198 . . 3 ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd𝐹) Fn 𝐴)
83, 6, 7sylancr 406 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹) Fn 𝐴)
9 rnco 4950 . . 3 ran (2nd𝐹) = ran (2nd ↾ ran 𝐹)
10 frn 5182 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4753 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12 rnss 4678 . . . . 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 5945 . . . . 5 (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15 frn 5182 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶)
1614, 15ax-mp 7 . . . 4 ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶
1713, 16syl6ss 3038 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
189, 17syl5eqss 3071 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd𝐹) ⊆ 𝐶)
19 df-f 5032 . 2 ((2nd𝐹):𝐴𝐶 ↔ ((2nd𝐹) Fn 𝐴 ∧ ran (2nd𝐹) ⊆ 𝐶))
208, 18, 19sylanbrc 409 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2620  wss 3000   × cxp 4450  ran crn 4453  cres 4454  ccom 4456   Fn wfn 5023  wf 5024  ontowfo 5026  2nd c2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034  df-fv 5036  df-2nd 5926
This theorem is referenced by: (None)
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