MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndcof Structured version   Visualization version   GIF version

Theorem 2ndcof 7709
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 7699 . . . 4 2nd :V–onto→V
2 fofn 6585 . . . 4 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . 3 2nd Fn V
4 ffn 6507 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 6509 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 219 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 6536 . . 3 ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd𝐹) Fn 𝐴)
83, 6, 7sylancr 587 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹) Fn 𝐴)
9 rnco 6098 . . 3 ran (2nd𝐹) = ran (2nd ↾ ran 𝐹)
10 frn 6513 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 5874 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12 rnss 5802 . . . . 5 ((2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
1310, 11, 123syl 18 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
14 f2ndres 7703 . . . . 5 (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15 frn 6513 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶)
1614, 15ax-mp 5 . . . 4 ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶
1713, 16sstrdi 3976 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
189, 17eqsstrid 4012 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd𝐹) ⊆ 𝐶)
19 df-f 6352 . 2 ((2nd𝐹):𝐴𝐶 ↔ ((2nd𝐹) Fn 𝐴 ∧ ran (2nd𝐹) ⊆ 𝐶))
208, 18, 19sylanbrc 583 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3492  wss 3933   × cxp 5546  ran crn 5549  cres 5550  ccom 5552   Fn wfn 6343  wf 6344  ontowfo 6346  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-2nd 7679
This theorem is referenced by:  axdc4lem  9865
  Copyright terms: Public domain W3C validator