Theorem caofidlcan 7735

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 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7104	. . . . . 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 3202	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))
8		oveq1 7438	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
9	8	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦) = 𝑦 ↔ ((𝐹‘𝑤)𝑅𝑦) = 𝑦))
10		eqeq1 2741	. . . . . . . 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 7439	. . . . . . . . 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 2753	. . . . . . . 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 3633	. . . . . 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 3176	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
20		ovexd 7466	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
21	20	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
22		mpteqb 7035	. . . 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 3146	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) ∈ 𝑆)
25		mpteqb 7035	. . . 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 6977	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	3	feqmptd 6977	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	28, 2, 4, 29, 30	offval2 7717	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	31, 30	eqeq12d 2753	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))))
33		fconstmpt 5747	. . . 4 ⊢ (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 )
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 ))
35	29, 34	eqeq12d 2753	. 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 2108  ∀wral 3061  Vcvv 3480  {csn 4626   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697

This theorem is referenced by:  psdmvr  22173