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Theorem fcomptf 29798
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 6543. (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 2913 . . . . 5 𝑥𝐴
2 nfcv 2913 . . . . 5 𝑥𝐷
3 nfcv 2913 . . . . 5 𝑥𝐸
41, 2, 3nff 6181 . . . 4 𝑥 𝐴:𝐷𝐸
5 fcomptf.1 . . . . 5 𝑥𝐵
6 nfcv 2913 . . . . 5 𝑥𝐶
75, 6, 2nff 6181 . . . 4 𝑥 𝐵:𝐶𝐷
84, 7nfan 1980 . . 3 𝑥(𝐴:𝐷𝐸𝐵:𝐶𝐷)
9 ffvelrn 6500 . . . . 5 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
109adantll 693 . . . 4 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
1110ex 397 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝑥𝐶 → (𝐵𝑥) ∈ 𝐷))
128, 11ralrimi 3106 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → ∀𝑥𝐶 (𝐵𝑥) ∈ 𝐷)
13 ffn 6185 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
1413adantl 467 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
155dffn5f 6394 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
1614, 15sylib 208 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
17 ffn 6185 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
1817adantr 466 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
19 dffn5 6383 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
2018, 19sylib 208 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
21 fveq2 6332 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
2212, 16, 20, 21fmptcof 6540 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wnfc 2900  cmpt 4863  ccom 5253   Fn wfn 6026  wf 6027  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  ofoprabco  29804
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