Theorem fcomptf 30995

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 7005. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Hypothesis

Ref	Expression
fcomptf.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
fcomptf	⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

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.

Step	Hyp	Ref	Expression
1		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐴
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐷
3		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐸
4	1, 2, 3	nff 6596	. . . 4 ⊢ Ⅎ𝑥 𝐴:𝐷⟶𝐸
5		fcomptf.1	. . . . 5 ⊢ Ⅎ𝑥𝐵
6		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6, 2	nff 6596	. . . 4 ⊢ Ⅎ𝑥 𝐵:𝐶⟶𝐷
8	4, 7	nfan 1902	. . 3 ⊢ Ⅎ𝑥(𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷)
9		ffvelrn 6959	. . . . 5 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
10	9	adantll 711	. . . 4 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
11	10	ex 413	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝑥 ∈ 𝐶 → (𝐵‘𝑥) ∈ 𝐷))
12	8, 11	ralrimi 3141	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → ∀𝑥 ∈ 𝐶 (𝐵‘𝑥) ∈ 𝐷)
13		ffn 6600	. . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶)
14	13	adantl 482	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶)
15	5	dffn5f 6840	. . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
16	14, 15	sylib 217	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
17		ffn 6600	. . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷)
18	17	adantr 481	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷)
19		dffn5 6828	. . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
20	18, 19	sylib 217	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
21		fveq2 6774	. 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥)))
22	12, 16, 20, 21	fmptcof 7002	1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887   ↦ cmpt 5157   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  ofoprabco  31001