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Theorem fcomptf 32387
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 7126. (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 2897 . . . . 5 𝑥𝐴
2 nfcv 2897 . . . . 5 𝑥𝐷
3 nfcv 2897 . . . . 5 𝑥𝐸
41, 2, 3nff 6706 . . . 4 𝑥 𝐴:𝐷𝐸
5 fcomptf.1 . . . . 5 𝑥𝐵
6 nfcv 2897 . . . . 5 𝑥𝐶
75, 6, 2nff 6706 . . . 4 𝑥 𝐵:𝐶𝐷
84, 7nfan 1894 . . 3 𝑥(𝐴:𝐷𝐸𝐵:𝐶𝐷)
9 ffvelcdm 7076 . . . . 5 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
109adantll 711 . . . 4 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
1110ex 412 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝑥𝐶 → (𝐵𝑥) ∈ 𝐷))
128, 11ralrimi 3248 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → ∀𝑥𝐶 (𝐵𝑥) ∈ 𝐷)
13 ffn 6710 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
1413adantl 481 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
155dffn5f 6956 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
1614, 15sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
17 ffn 6710 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
1817adantr 480 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
19 dffn5 6943 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
2018, 19sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
21 fveq2 6884 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
2212, 16, 20, 21fmptcof 7123 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wnfc 2877  cmpt 5224  ccom 5673   Fn wfn 6531  wf 6532  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by:  ofoprabco  32393
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