Theorem caofinvl 7700

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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑁:𝑆⟶𝑆)
5		caofref.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6	5	ffvelcdmda 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑆)
7	4, 6	ffvelcdmd 7088	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑁‘(𝐹‘𝑣)) ∈ 𝑆)
8	2, 7	fmpt3d 7116	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
9	8	ffvelcdmda 7087	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
10	5	ffvelcdmda 7087	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
11		fvex 6905	. . . . . . 7 ⊢ (𝑁‘(𝐹‘𝑣)) ∈ V
12		eqid 2733	. . . . . . 7 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))
13	11, 12	fnmpti 6694	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴
14	2	fneq1d 6643	. . . . . 6 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴))
15	13, 14	mpbiri 258	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
16		dffn5 6951	. . . . 5 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
17	15, 16	sylib 217	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
18	5	feqmptd 6961	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
19	1, 9, 10, 17, 18	offval2 7690	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))))
20	2	fveq1d 6894	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤))
21		2fveq3 6897	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑁‘(𝐹‘𝑣)) = (𝑁‘(𝐹‘𝑤)))
22		fvex 6905	. . . . . . . 8 ⊢ (𝑁‘(𝐹‘𝑤)) ∈ V
23	21, 12, 22	fvmpt 6999	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤)))
24	20, 23	sylan9eq 2793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝑁‘(𝐹‘𝑤)))
25	24	oveq1d 7424	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
26		fveq2 6892	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑁‘𝑥) = (𝑁‘(𝐹‘𝑤)))
27		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
28	26, 27	oveq12d 7427	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑁‘𝑥)𝑅𝑥) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
29	28	eqeq1d 2735	. . . . . 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 3147	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
32	31	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
33	29, 32, 10	rspcdva 3614	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵)
34	25, 33	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = 𝐵)
35	34	mpteq2dva 5249	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ 𝐵))
36	19, 35	eqtrd 2773	. 2 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ 𝐵))
37		fconstmpt 5739	. 2 ⊢ (𝐴 × {𝐵}) = (𝑤 ∈ 𝐴 ↦ 𝐵)
38	36, 37	eqtr4di 2791	1 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {csn 4629   ↦ cmpt 5232   × cxp 5675   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ∘_f cof 7668

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670

This theorem is referenced by:  grpvlinv  21897  lflnegl  37946