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Theorem fcomptf 32668
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 7153. (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 2905 . . . . 5 𝑥𝐴
2 nfcv 2905 . . . . 5 𝑥𝐷
3 nfcv 2905 . . . . 5 𝑥𝐸
41, 2, 3nff 6732 . . . 4 𝑥 𝐴:𝐷𝐸
5 fcomptf.1 . . . . 5 𝑥𝐵
6 nfcv 2905 . . . . 5 𝑥𝐶
75, 6, 2nff 6732 . . . 4 𝑥 𝐵:𝐶𝐷
84, 7nfan 1899 . . 3 𝑥(𝐴:𝐷𝐸𝐵:𝐶𝐷)
9 ffvelcdm 7101 . . . . 5 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
109adantll 714 . . . 4 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
1110ex 412 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝑥𝐶 → (𝐵𝑥) ∈ 𝐷))
128, 11ralrimi 3257 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → ∀𝑥𝐶 (𝐵𝑥) ∈ 𝐷)
13 ffn 6736 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
1413adantl 481 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
155dffn5f 6980 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
1614, 15sylib 218 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
17 ffn 6736 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
1817adantr 480 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
19 dffn5 6967 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
2018, 19sylib 218 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
21 fveq2 6906 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
2212, 16, 20, 21fmptcof 7150 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wnfc 2890  cmpt 5225  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  ofoprabco  32674
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