Theorem offval2f 7420

Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)

Hypotheses

Ref	Expression
offval2f.0	⊢ Ⅎ𝑥𝜑
offval2f.a	⊢ Ⅎ𝑥𝐴
offval2f.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval2f.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
offval2f.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
offval2f.4	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
offval2f.5	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))

Assertion

Ref	Expression
offval2f	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Distinct variable group:   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem offval2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2f.0	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2		offval2f.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
3	2	ex 415	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊))
4	1, 3	ralrimi 3216	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
5		offval2f.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
6	5	fnmptf 6483	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
7	4, 6	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8		offval2f.4	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8	fneq1d 6445	. . . 4 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
10	7, 9	mpbird 259	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11		offval2f.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
12	11	ex 415	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋))
13	1, 12	ralrimi 3216	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋)
14	5	fnmptf 6483	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
16		offval2f.5	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
17	16	fneq1d 6445	. . . 4 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
18	15, 17	mpbird 259	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
19		offval2f.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		inidm 4194	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21	8	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	fveq1d 6671	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
23	16	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
24	23	fveq1d 6671	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
25	10, 18, 19, 19, 20, 22, 24	offval 7415	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))))
26		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦𝐴
27		nffvmpt1 6680	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
28		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥𝑅
29		nffvmpt1 6680	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
30	27, 28, 29	nfov 7185	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
31		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
32		fveq2 6669	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
33		fveq2 6669	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
34	32, 33	oveq12d 7173	. . . 4 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)) = (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)))
35	26, 5, 30, 31, 34	cbvmptf 5164	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))) = (𝑥 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)))
36		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
37	5	fvmpt2f 6768	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
38	36, 2, 37	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
39	5	fvmpt2f 6768	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
40	36, 11, 39	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
41	38, 40	oveq12d 7173	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)) = (𝐵𝑅𝐶))
42	1, 41	mpteq2da 5159	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
43	35, 42	syl5eq 2868	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
44	25, 43	eqtrd 2856	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961  ∀wral 3138   ↦ cmpt 5145   Fn wfn 6349  ‘cfv 6354  (class class class)co 7155   ∘_f cof 7406

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408

This theorem is referenced by:  esumaddf  31320  binomcxplemnotnn0  40686