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Theorem 1stcof 5916
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 5910 . . . 4 1st :V–onto→V
2 fofn 5219 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 7 . . 3 1st Fn V
4 ffn 5147 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5149 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 120 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5170 . . 3 ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st𝐹) Fn 𝐴)
83, 6, 7sylancr 405 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹) Fn 𝐴)
9 rnco 4924 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 5155 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4727 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)))
12 rnss 4653 . . . . 5 ((1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶)))
1310, 11, 123syl 17 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶)))
14 f1stres 5912 . . . . 5 (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15 frn 5155 . . . . 5 ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
1614, 15ax-mp 7 . . . 4 ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
1713, 16syl6ss 3035 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵)
189, 17syl5eqss 3068 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st𝐹) ⊆ 𝐵)
19 df-f 5006 . 2 ((1st𝐹):𝐴𝐵 ↔ ((1st𝐹) Fn 𝐴 ∧ ran (1st𝐹) ⊆ 𝐵))
208, 18, 19sylanbrc 408 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2619  wss 2997   × cxp 4426  ran crn 4429  cres 4430  ccom 4432   Fn wfn 4997  wf 4998  ontowfo 5000  1st c1st 5891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893
This theorem is referenced by: (None)
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