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Theorem 2ndcof 6030
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 6024 . . . 4 2nd :V–onto→V
2 fofn 5317 . . . 4 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . 3 2nd Fn V
4 ffn 5242 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5244 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 121 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5267 . . 3 ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd𝐹) Fn 𝐴)
83, 6, 7sylancr 410 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹) Fn 𝐴)
9 rnco 5015 . . 3 ran (2nd𝐹) = ran (2nd ↾ ran 𝐹)
10 frn 5251 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4816 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12 rnss 4739 . . . . 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 6026 . . . . 5 (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15 frn 5251 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶)
1614, 15ax-mp 5 . . . 4 ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶
1713, 16sstrdi 3079 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
189, 17eqsstrid 3113 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd𝐹) ⊆ 𝐶)
19 df-f 5097 . 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 2660  wss 3041   × cxp 4507  ran crn 4510  cres 4511  ccom 4513   Fn wfn 5088  wf 5089  ontowfo 5091  2nd c2nd 6005
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-2nd 6007
This theorem is referenced by: (None)
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