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Theorem 2ndcof 8010
Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
2ndcof (š¹:š“āŸ¶(šµ Ɨ š¶) → (2nd ∘ š¹):š“āŸ¶š¶)

Proof of Theorem 2ndcof
StepHypRef Expression
1 fo2nd 8000 . . . 4 2nd :V–onto→V
2 fofn 6807 . . . 4 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . 3 2nd Fn V
4 ffn 6717 . . . 4 (š¹:š“āŸ¶(šµ Ɨ š¶) → š¹ Fn š“)
5 dffn2 6719 . . . 4 (š¹ Fn š“ ↔ š¹:š“āŸ¶V)
64, 5sylib 217 . . 3 (š¹:š“āŸ¶(šµ Ɨ š¶) → š¹:š“āŸ¶V)
7 fnfco 6756 . . 3 ((2nd Fn V ∧ š¹:š“āŸ¶V) → (2nd ∘ š¹) Fn š“)
83, 6, 7sylancr 586 . 2 (š¹:š“āŸ¶(šµ Ɨ š¶) → (2nd ∘ š¹) Fn š“)
9 rnco 6251 . . 3 ran (2nd ∘ š¹) = ran (2nd ↾ ran š¹)
10 frn 6724 . . . . 5 (š¹:š“āŸ¶(šµ Ɨ š¶) → ran š¹ āŠ† (šµ Ɨ š¶))
11 ssres2 6009 . . . . 5 (ran š¹ āŠ† (šµ Ɨ š¶) → (2nd ↾ ran š¹) āŠ† (2nd ↾ (šµ Ɨ š¶)))
12 rnss 5938 . . . . 5 ((2nd ↾ ran š¹) āŠ† (2nd ↾ (šµ Ɨ š¶)) → ran (2nd ↾ ran š¹) āŠ† ran (2nd ↾ (šµ Ɨ š¶)))
1310, 11, 123syl 18 . . . 4 (š¹:š“āŸ¶(šµ Ɨ š¶) → ran (2nd ↾ ran š¹) āŠ† ran (2nd ↾ (šµ Ɨ š¶)))
14 f2ndres 8004 . . . . 5 (2nd ↾ (šµ Ɨ š¶)):(šµ Ɨ š¶)āŸ¶š¶
15 frn 6724 . . . . 5 ((2nd ↾ (šµ Ɨ š¶)):(šµ Ɨ š¶)āŸ¶š¶ → ran (2nd ↾ (šµ Ɨ š¶)) āŠ† š¶)
1614, 15ax-mp 5 . . . 4 ran (2nd ↾ (šµ Ɨ š¶)) āŠ† š¶
1713, 16sstrdi 3994 . . 3 (š¹:š“āŸ¶(šµ Ɨ š¶) → ran (2nd ↾ ran š¹) āŠ† š¶)
189, 17eqsstrid 4030 . 2 (š¹:š“āŸ¶(šµ Ɨ š¶) → ran (2nd ∘ š¹) āŠ† š¶)
19 df-f 6547 . 2 ((2nd ∘ š¹):š“āŸ¶š¶ ↔ ((2nd ∘ š¹) Fn š“ ∧ ran (2nd ∘ š¹) āŠ† š¶))
208, 18, 19sylanbrc 582 1 (š¹:š“āŸ¶(šµ Ɨ š¶) → (2nd ∘ š¹):š“āŸ¶š¶)
Colors of variables: wff setvar class
Syntax hints:   → wi 4  Vcvv 3473   āŠ† wss 3948   Ɨ cxp 5674  ran crn 5677   ↾ cres 5678   ∘ ccom 5680   Fn wfn 6538  āŸ¶wf 6539  ā€“onto→wfo 6541  2nd c2nd 7978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-2nd 7980
This theorem is referenced by:  axdc4lem  10456
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