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Theorem 1stcof 8043
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
Assertion
Ref Expression
1stcof (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)

Proof of Theorem 1stcof
StepHypRef Expression
1 fo1st 8033 . . . 4 1st :V–onto→V
2 fofn 6823 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . 3 1st Fn V
4 ffn 6737 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 6739 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 218 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 6774 . . 3 ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st𝐹) Fn 𝐴)
83, 6, 7sylancr 587 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹) Fn 𝐴)
9 rnco 6274 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 6744 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 6025 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)))
12 rnss 5953 . . . . 5 ((1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶)))
1310, 11, 123syl 18 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶)))
14 f1stres 8037 . . . . 5 (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15 frn 6744 . . . . 5 ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
1614, 15ax-mp 5 . . . 4 ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
1713, 16sstrdi 4008 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵)
189, 17eqsstrid 4044 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st𝐹) ⊆ 𝐵)
19 df-f 6567 . 2 ((1st𝐹):𝐴𝐵 ↔ ((1st𝐹) Fn 𝐴 ∧ ran (1st𝐹) ⊆ 𝐵))
208, 18, 19sylanbrc 583 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3478  wss 3963   × cxp 5687  ran crn 5690  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  ontowfo 6561  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-1st 8013
This theorem is referenced by:  ruclem11  16273  ruclem12  16274  caubl  25356
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