Theorem caofinvl 7563

Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofinv.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)
caofinv.4	⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
caofinv.5	⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))
caofinvl.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)

Assertion

Ref	Expression
caofinvl	⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵}))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑣,𝐴   𝑣,𝐹,𝑥   𝑥,𝑁,𝑣   𝑣,𝑆   𝜑,𝑣

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑣)   𝑅(𝑣)   𝐺(𝑣)   𝑉(𝑥,𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem caofinvl

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		caofinv.5	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))
3		caofinv.4	. . . . . . . 8 ⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑁:𝑆⟶𝑆)
5		caofref.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6	5	ffvelrnda 6961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑆)
7	4, 6	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑁‘(𝐹‘𝑣)) ∈ 𝑆)
8	2, 7	fmpt3d 6990	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
9	8	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
10	5	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
11		fvex 6787	. . . . . . 7 ⊢ (𝑁‘(𝐹‘𝑣)) ∈ V
12		eqid 2738	. . . . . . 7 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))
13	11, 12	fnmpti 6576	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴
14	2	fneq1d 6526	. . . . . 6 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴))
15	13, 14	mpbiri 257	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
16		dffn5 6828	. . . . 5 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
17	15, 16	sylib 217	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
18	5	feqmptd 6837	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
19	1, 9, 10, 17, 18	offval2 7553	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))))
20	2	fveq1d 6776	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤))
21		2fveq3 6779	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑁‘(𝐹‘𝑣)) = (𝑁‘(𝐹‘𝑤)))
22		fvex 6787	. . . . . . . 8 ⊢ (𝑁‘(𝐹‘𝑤)) ∈ V
23	21, 12, 22	fvmpt 6875	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤)))
24	20, 23	sylan9eq 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝑁‘(𝐹‘𝑤)))
25	24	oveq1d 7290	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
26		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑁‘𝑥) = (𝑁‘(𝐹‘𝑤)))
27		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
28	26, 27	oveq12d 7293	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑁‘𝑥)𝑅𝑥) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
29	28	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑁‘𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵))
30		caofinvl.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
31	30	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
32	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
33	29, 32, 10	rspcdva 3562	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵)
34	25, 33	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = 𝐵)
35	34	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ 𝐵))
36	19, 35	eqtrd 2778	. 2 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ 𝐵))
37		fconstmpt 5649	. 2 ⊢ (𝐴 × {𝐵}) = (𝑤 ∈ 𝐴 ↦ 𝐵)
38	36, 37	eqtr4di 2796	1 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {csn 4561   ↦ cmpt 5157   × cxp 5587   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531

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-rep 5209  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-reu 3072  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-iun 4926  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-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533

This theorem is referenced by:  grpvlinv  21544  lflnegl  37090