Theorem fcomptf 32731

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 7086. (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 2898	. . . . 5 ⊢ Ⅎ𝑥𝐴
2		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐷
3		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐸
4	1, 2, 3	nff 6664	. . . 4 ⊢ Ⅎ𝑥 𝐴:𝐷⟶𝐸
5		fcomptf.1	. . . . 5 ⊢ Ⅎ𝑥𝐵
6		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6, 2	nff 6664	. . . 4 ⊢ Ⅎ𝑥 𝐵:𝐶⟶𝐷
8	4, 7	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷)
9		ffvelcdm 7033	. . . . 5 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
10	9	adantll 715	. . . 4 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
11	10	ex 412	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝑥 ∈ 𝐶 → (𝐵‘𝑥) ∈ 𝐷))
12	8, 11	ralrimi 3235	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → ∀𝑥 ∈ 𝐶 (𝐵‘𝑥) ∈ 𝐷)
13		ffn 6668	. . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶)
14	13	adantl 481	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶)
15	5	dffn5f 6911	. . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
16	14, 15	sylib 218	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
17		ffn 6668	. . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷)
18	17	adantr 480	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷)
19		dffn5 6898	. . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
20	18, 19	sylib 218	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
21		fveq2 6840	. 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥)))
22	12, 16, 20, 21	fmptcof 7083	1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2883   ↦ cmpt 5166   ∘ ccom 5635   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  ofoprabco  32737