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Theorem fcomptf 30331
Description: Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6887. (Contributed by Thierry Arnoux, 30-Jun-2017.)
Hypothesis
Ref Expression
fcomptf.1 𝑥𝐵
Assertion
Ref Expression
fcomptf ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fcomptf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2974 . . . . 5 𝑥𝐴
2 nfcv 2974 . . . . 5 𝑥𝐷
3 nfcv 2974 . . . . 5 𝑥𝐸
41, 2, 3nff 6503 . . . 4 𝑥 𝐴:𝐷𝐸
5 fcomptf.1 . . . . 5 𝑥𝐵
6 nfcv 2974 . . . . 5 𝑥𝐶
75, 6, 2nff 6503 . . . 4 𝑥 𝐵:𝐶𝐷
84, 7nfan 1891 . . 3 𝑥(𝐴:𝐷𝐸𝐵:𝐶𝐷)
9 ffvelrn 6841 . . . . 5 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
109adantll 710 . . . 4 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
1110ex 413 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝑥𝐶 → (𝐵𝑥) ∈ 𝐷))
128, 11ralrimi 3213 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → ∀𝑥𝐶 (𝐵𝑥) ∈ 𝐷)
13 ffn 6507 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
1413adantl 482 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
155dffn5f 6729 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
1614, 15sylib 219 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
17 ffn 6507 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
1817adantr 481 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
19 dffn5 6717 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
2018, 19sylib 219 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
21 fveq2 6663 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
2212, 16, 20, 21fmptcof 6884 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wnfc 2958  cmpt 5137  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  ofoprabco  30337
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