Theorem caofidlcan 7651

Description: Transfer a cancellation/identity law to the function operation. (Contributed by SN, 16-Oct-2025.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofcom.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofidlcan.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))

Assertion

Ref	Expression
caofidlcan	⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ 𝐹 = (𝐴 × { 0 })))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥, 0 ,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caofidlcan

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.2	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2	1	ffvelcdmda 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5	2, 4	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆))
6		caofidlcan.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))
7	6	ralrimivva 3172	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))
8		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
9	8	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦) = 𝑦 ↔ ((𝐹‘𝑤)𝑅𝑦) = 𝑦))
10		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥 = 0 ↔ (𝐹‘𝑤) = 0 ))
11	9, 10	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ) ↔ (((𝐹‘𝑤)𝑅𝑦) = 𝑦 ↔ (𝐹‘𝑤) = 0 )))
12		oveq2 7357	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
13		id 22	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑤) → 𝑦 = (𝐺‘𝑤))
14	12, 13	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦) = 𝑦 ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤)))
15	14	bibi1d 343	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → ((((𝐹‘𝑤)𝑅𝑦) = 𝑦 ↔ (𝐹‘𝑤) = 0 ) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ (𝐹‘𝑤) = 0 )))
16	11, 15	rspc2v 3588	. . . . . 6 ⊢ (((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ (𝐹‘𝑤) = 0 )))
17	7, 16	mpan9 506	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ (𝐹‘𝑤) = 0 ))
18	5, 17	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ (𝐹‘𝑤) = 0 ))
19	18	ralbidva 3150	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
20		ovexd 7384	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
21	20	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
22		mpteqb 6949	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V → ((𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤)))
23	21, 22	syl 17	. . 3 ⊢ (𝜑 → ((𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤)))
24	2	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) ∈ 𝑆)
25		mpteqb 6949	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 (𝐹‘𝑤) ∈ 𝑆 → ((𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)) = (𝑤 ∈ 𝐴 ↦ 0 ) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
26	24, 25	syl 17	. . 3 ⊢ (𝜑 → ((𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)) = (𝑤 ∈ 𝐴 ↦ 0 ) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
27	19, 23, 26	3bitr4d 311	. 2 ⊢ (𝜑 → ((𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ↔ (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)) = (𝑤 ∈ 𝐴 ↦ 0 )))
28		caofref.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
29	1	feqmptd 6891	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	3	feqmptd 6891	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	28, 2, 4, 29, 30	offval2 7633	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	31, 30	eqeq12d 2745	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))))
33		fconstmpt 5681	. . . 4 ⊢ (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 )
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 ))
35	29, 34	eqeq12d 2745	. 2 ⊢ (𝜑 → (𝐹 = (𝐴 × { 0 }) ↔ (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)) = (𝑤 ∈ 𝐴 ↦ 0 )))
36	27, 32, 35	3bitr4d 311	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ 𝐹 = (𝐴 × { 0 })))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436  {csn 4577   ↦ cmpt 5173   × cxp 5617  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∘_f cof 7611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613

This theorem is referenced by:  psdmvr  22054