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Theorem fcompt 5276
Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcompt ((A:𝐷𝐸 B:𝐶𝐷) → (AB) = (x 𝐶 ↦ (A‘(Bx))))
Distinct variable groups:   x,A   x,B   x,𝐶   x,𝐷   x,𝐸

Proof of Theorem fcompt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 5243 . . 3 ((B:𝐶𝐷 x 𝐶) → (Bx) 𝐷)
21adantll 445 . 2 (((A:𝐷𝐸 B:𝐶𝐷) x 𝐶) → (Bx) 𝐷)
3 ffn 4989 . . . 4 (B:𝐶𝐷B Fn 𝐶)
43adantl 262 . . 3 ((A:𝐷𝐸 B:𝐶𝐷) → B Fn 𝐶)
5 dffn5im 5162 . . 3 (B Fn 𝐶B = (x 𝐶 ↦ (Bx)))
64, 5syl 14 . 2 ((A:𝐷𝐸 B:𝐶𝐷) → B = (x 𝐶 ↦ (Bx)))
7 ffn 4989 . . . 4 (A:𝐷𝐸A Fn 𝐷)
87adantr 261 . . 3 ((A:𝐷𝐸 B:𝐶𝐷) → A Fn 𝐷)
9 dffn5im 5162 . . 3 (A Fn 𝐷A = (y 𝐷 ↦ (Ay)))
108, 9syl 14 . 2 ((A:𝐷𝐸 B:𝐶𝐷) → A = (y 𝐷 ↦ (Ay)))
11 fveq2 5121 . 2 (y = (Bx) → (Ay) = (A‘(Bx)))
122, 6, 10, 11fmptco 5273 1 ((A:𝐷𝐸 B:𝐶𝐷) → (AB) = (x 𝐶 ↦ (A‘(Bx))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cmpt 3809  ccom 4292   Fn wfn 4840  wf 4841  cfv 4845
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-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
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-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-fv 4853
This theorem is referenced by: (None)
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