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Theorem fcompt 5558
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:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (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 5521 . . 3 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
21adantll 467 . 2 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
3 ffn 5242 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
43adantl 275 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
5 dffn5im 5435 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
64, 5syl 14 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
7 ffn 5242 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
87adantr 274 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
9 dffn5im 5435 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
108, 9syl 14 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
11 fveq2 5389 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
122, 6, 10, 11fmptco 5554 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cmpt 3959  ccom 4513   Fn wfn 5088  wf 5089  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by: (None)
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