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Theorem 1stcof 5732
 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 5726 . . . 4 1st :V–onto→V
2 fofn 5051 . . . 4 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 7 . . 3 1st Fn V
4 ffn 4989 . . . 4 (𝐹:A⟶(B × 𝐶) → 𝐹 Fn A)
5 dffn2 4990 . . . 4 (𝐹 Fn A𝐹:A⟶V)
64, 5sylib 127 . . 3 (𝐹:A⟶(B × 𝐶) → 𝐹:A⟶V)
7 fnfco 5008 . . 3 ((1st Fn V 𝐹:A⟶V) → (1st𝐹) Fn A)
83, 6, 7sylancr 393 . 2 (𝐹:A⟶(B × 𝐶) → (1st𝐹) Fn A)
9 rnco 4770 . . 3 ran (1st𝐹) = ran (1st ↾ ran 𝐹)
10 frn 4995 . . . . 5 (𝐹:A⟶(B × 𝐶) → ran 𝐹 ⊆ (B × 𝐶))
11 ssres2 4581 . . . . 5 (ran 𝐹 ⊆ (B × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (B × 𝐶)))
12 rnss 4507 . . . . 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 5728 . . . . 5 (1st ↾ (B × 𝐶)):(B × 𝐶)⟶B
15 frn 4995 . . . . 5 ((1st ↾ (B × 𝐶)):(B × 𝐶)⟶B → ran (1st ↾ (B × 𝐶)) ⊆ B)
1614, 15ax-mp 7 . . . 4 ran (1st ↾ (B × 𝐶)) ⊆ B
1713, 16syl6ss 2951 . . 3 (𝐹:A⟶(B × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ B)
189, 17syl5eqss 2983 . 2 (𝐹:A⟶(B × 𝐶) → ran (1st𝐹) ⊆ B)
19 df-f 4849 . 2 ((1st𝐹):AB ↔ ((1st𝐹) Fn A ran (1st𝐹) ⊆ B))
208, 18, 19sylanbrc 394 1 (𝐹:A⟶(B × 𝐶) → (1st𝐹):AB)
 Colors of variables: wff set class Syntax hints:   → wi 4  Vcvv 2551   ⊆ wss 2911   × cxp 4286  ran crn 4289   ↾ cres 4290   ∘ ccom 4292   Fn wfn 4840  ⟶wf 4841  –onto→wfo 4843  1st c1st 5707 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709 This theorem is referenced by: (None)
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