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Theorem fcomptf 32675
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 2903 . . . . 5 𝑥𝐴
2 nfcv 2903 . . . . 5 𝑥𝐷
3 nfcv 2903 . . . . 5 𝑥𝐸
41, 2, 3nff 6733 . . . 4 𝑥 𝐴:𝐷𝐸
5 fcomptf.1 . . . . 5 𝑥𝐵
6 nfcv 2903 . . . . 5 𝑥𝐶
75, 6, 2nff 6733 . . . 4 𝑥 𝐵:𝐶𝐷
84, 7nfan 1897 . . 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 3255 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → ∀𝑥𝐶 (𝐵𝑥) ∈ 𝐷)
13 ffn 6737 . . . 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 6737 . . . 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 6907 . 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 1537  wcel 2106  wnfc 2888  cmpt 5231  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  ofoprabco  32681
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