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Theorem 1stcof 6111
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 6105 . . . 4 1st :V–onto→V
2 fofn 5394 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . 3 1st Fn V
4 ffn 5319 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5321 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 121 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5344 . . 3 ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st𝐹) Fn 𝐴)
83, 6, 7sylancr 411 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹) Fn 𝐴)
9 rnco 5092 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 5328 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4893 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)))
12 rnss 4816 . . . . 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 6107 . . . . 5 (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15 frn 5328 . . . . 5 ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
1614, 15ax-mp 5 . . . 4 ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
1713, 16sstrdi 3140 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵)
189, 17eqsstrid 3174 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st𝐹) ⊆ 𝐵)
19 df-f 5174 . 2 ((1st𝐹):𝐴𝐵 ↔ ((1st𝐹) Fn 𝐴 ∧ ran (1st𝐹) ⊆ 𝐵))
208, 18, 19sylanbrc 414 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2712  wss 3102   × cxp 4584  ran crn 4587  cres 4588  ccom 4590   Fn wfn 5165  wf 5166  ontowfo 5168  1st c1st 6086
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176  df-fv 5178  df-1st 6088
This theorem is referenced by: (None)
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