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Theorem 1stcof 6142
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 6136 . . . 4 1st :V–onto→V
2 fofn 5422 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . 3 1st Fn V
4 ffn 5347 . . . 4 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5 dffn2 5349 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
64, 5sylib 121 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7 fnfco 5372 . . 3 ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st𝐹) Fn 𝐴)
83, 6, 7sylancr 412 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹) Fn 𝐴)
9 rnco 5117 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 5356 . . . . 5 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11 ssres2 4918 . . . . 5 (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)))
12 rnss 4841 . . . . 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 6138 . . . . 5 (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15 frn 5356 . . . . 5 ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
1614, 15ax-mp 5 . . . 4 ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
1713, 16sstrdi 3159 . . 3 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵)
189, 17eqsstrid 3193 . 2 (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st𝐹) ⊆ 𝐵)
19 df-f 5202 . 2 ((1st𝐹):𝐴𝐵 ↔ ((1st𝐹) Fn 𝐴 ∧ ran (1st𝐹) ⊆ 𝐵))
208, 18, 19sylanbrc 415 1 (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2730  wss 3121   × cxp 4609  ran crn 4612  cres 4613  ccom 4615   Fn wfn 5193  wf 5194  ontowfo 5196  1st c1st 6117
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119
This theorem is referenced by: (None)
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