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Theorem 2ndcof 7953
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¶) ā†’ (2nd āˆ˜ š¹):š“āŸ¶š¶)

Proof of Theorem 2ndcof
StepHypRef Expression
1 fo2nd 7943 . . . 4 2nd :Vā€“ontoā†’V
2 fofn 6759 . . . 4 (2nd :Vā€“ontoā†’V ā†’ 2nd Fn V)
31, 2ax-mp 5 . . 3 2nd Fn V
4 ffn 6669 . . . 4 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ š¹ Fn š“)
5 dffn2 6671 . . . 4 (š¹ Fn š“ ā†” š¹:š“āŸ¶V)
64, 5sylib 217 . . 3 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ š¹:š“āŸ¶V)
7 fnfco 6708 . . 3 ((2nd Fn V āˆ§ š¹:š“āŸ¶V) ā†’ (2nd āˆ˜ š¹) Fn š“)
83, 6, 7sylancr 588 . 2 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ (2nd āˆ˜ š¹) Fn š“)
9 rnco 6205 . . 3 ran (2nd āˆ˜ š¹) = ran (2nd ā†¾ ran š¹)
10 frn 6676 . . . . 5 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ ran š¹ āŠ† (šµ Ɨ š¶))
11 ssres2 5966 . . . . 5 (ran š¹ āŠ† (šµ Ɨ š¶) ā†’ (2nd ā†¾ ran š¹) āŠ† (2nd ā†¾ (šµ Ɨ š¶)))
12 rnss 5895 . . . . 5 ((2nd ā†¾ ran š¹) āŠ† (2nd ā†¾ (šµ Ɨ š¶)) ā†’ ran (2nd ā†¾ ran š¹) āŠ† ran (2nd ā†¾ (šµ Ɨ š¶)))
1310, 11, 123syl 18 . . . 4 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ ran (2nd ā†¾ ran š¹) āŠ† ran (2nd ā†¾ (šµ Ɨ š¶)))
14 f2ndres 7947 . . . . 5 (2nd ā†¾ (šµ Ɨ š¶)):(šµ Ɨ š¶)āŸ¶š¶
15 frn 6676 . . . . 5 ((2nd ā†¾ (šµ Ɨ š¶)):(šµ Ɨ š¶)āŸ¶š¶ ā†’ ran (2nd ā†¾ (šµ Ɨ š¶)) āŠ† š¶)
1614, 15ax-mp 5 . . . 4 ran (2nd ā†¾ (šµ Ɨ š¶)) āŠ† š¶
1713, 16sstrdi 3957 . . 3 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ ran (2nd ā†¾ ran š¹) āŠ† š¶)
189, 17eqsstrid 3993 . 2 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ ran (2nd āˆ˜ š¹) āŠ† š¶)
19 df-f 6501 . 2 ((2nd āˆ˜ š¹):š“āŸ¶š¶ ā†” ((2nd āˆ˜ š¹) Fn š“ āˆ§ ran (2nd āˆ˜ š¹) āŠ† š¶))
208, 18, 19sylanbrc 584 1 (š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ (2nd āˆ˜ š¹):š“āŸ¶š¶)
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4  Vcvv 3446   āŠ† wss 3911   Ɨ cxp 5632  ran crn 5635   ā†¾ cres 5636   āˆ˜ ccom 5638   Fn wfn 6492  āŸ¶wf 6493  ā€“ontoā†’wfo 6495  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-2nd 7923
This theorem is referenced by:  axdc4lem  10392
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