Theorem caofidlcan 7707

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 7073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7073	. . . . . 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 3187	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))
8		oveq1 7410	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
9	8	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦) = 𝑦 ↔ ((𝐹‘𝑤)𝑅𝑦) = 𝑦))
10		eqeq1 2739	. . . . . . . 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 7411	. . . . . . . . 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 2751	. . . . . . . 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 3612	. . . . . 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 3161	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
20		ovexd 7438	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
21	20	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
22		mpteqb 7004	. . . 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 3132	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) ∈ 𝑆)
25		mpteqb 7004	. . . 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 6946	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	3	feqmptd 6946	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	28, 2, 4, 29, 30	offval2 7689	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	31, 30	eqeq12d 2751	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))))
33		fconstmpt 5716	. . . 4 ⊢ (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 )
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 ))
35	29, 34	eqeq12d 2751	. 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 3051  Vcvv 3459  {csn 4601   ↦ cmpt 5201   × cxp 5652  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403   ∘_f cof 7667

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669

This theorem is referenced by:  psdmvr  22105