Theorem caofinvl 6250

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.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
3	2	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑁:𝑆⟶𝑆)
4		caofref.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
5	4	ffvelcdmda 5772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑆)
6	3, 5	ffvelcdmd 5773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑁‘(𝐹‘𝑣)) ∈ 𝑆)
7		eqid 2229	. . . . . . 7 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))
8	6, 7	fmptd 5791	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))):𝐴⟶𝑆)
9		caofinv.5	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))
10	9	feq1d 5460	. . . . . 6 ⊢ (𝜑 → (𝐺:𝐴⟶𝑆 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))):𝐴⟶𝑆))
11	8, 10	mpbird 167	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
12	11	ffvelcdmda 5772	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
13	4	ffvelcdmda 5772	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
14	6	ralrimiva 2603	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ 𝐴 (𝑁‘(𝐹‘𝑣)) ∈ 𝑆)
15	7	fnmpt 5450	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝐴 (𝑁‘(𝐹‘𝑣)) ∈ 𝑆 → (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴)
16	14, 15	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴)
17	9	fneq1d 5411	. . . . . 6 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴))
18	16, 17	mpbird 167	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
19		dffn5im 5681	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
21	4	feqmptd 5689	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
22	1, 12, 13, 20, 21	offval2 6240	. . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))))
23	9	fveq1d 5631	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤))
24	23	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤))
25		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
26	2	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑁:𝑆⟶𝑆)
27	26, 13	ffvelcdmd 5773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑁‘(𝐹‘𝑤)) ∈ 𝑆)
28		fveq2 5629	. . . . . . . . . 10 ⊢ (𝑣 = 𝑤 → (𝐹‘𝑣) = (𝐹‘𝑤))
29	28	fveq2d 5633	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (𝑁‘(𝐹‘𝑣)) = (𝑁‘(𝐹‘𝑤)))
30	29, 7	fvmptg 5712	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ (𝑁‘(𝐹‘𝑤)) ∈ 𝑆) → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤)))
31	25, 27, 30	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤)))
32	24, 31	eqtrd 2262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝑁‘(𝐹‘𝑤)))
33	32	oveq1d 6022	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
34		fveq2 5629	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑁‘𝑥) = (𝑁‘(𝐹‘𝑤)))
35		id 19	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
36	34, 35	oveq12d 6025	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑁‘𝑥)𝑅𝑥) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
37	36	eqeq1d 2238	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑁‘𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵))
38		caofinvl.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
39	38	ralrimiva 2603	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
40	39	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
41	37, 40, 13	rspcdva 2912	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵)
42	33, 41	eqtrd 2262	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = 𝐵)
43	42	mpteq2dva 4174	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ 𝐵))
44	22, 43	eqtrd 2262	. 2 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ 𝐵))
45		fconstmpt 4766	. 2 ⊢ (𝐴 × {𝐵}) = (𝑤 ∈ 𝐴 ↦ 𝐵)
46	44, 45	eqtr4di 2280	1 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {csn 3666   ↦ cmpt 4145   × cxp 4717   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ∘_𝑓 cof 6222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224

This theorem is referenced by: (None)