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Theorem 1stcof 8044
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 8034 . . . 4 1st :V–onto→V
2 fofn 6822 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . 3 1st Fn V
4 ffn 6736 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 6738 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 218 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 6773 . . 3 ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st𝐹) Fn 𝐴)
83, 6, 7sylancr 587 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹) Fn 𝐴)
9 rnco 6272 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 6743 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 6022 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)))
12 rnss 5950 . . . . 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 8038 . . . . 5 (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15 frn 6743 . . . . 5 ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
1614, 15ax-mp 5 . . . 4 ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
1713, 16sstrdi 3996 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵)
189, 17eqsstrid 4022 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st𝐹) ⊆ 𝐵)
19 df-f 6565 . 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 3480  wss 3951   × cxp 5683  ran crn 5686  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  ontowfo 6559  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-1st 8014
This theorem is referenced by:  ruclem11  16276  ruclem12  16277  caubl  25342
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