Theorem caofidlcan 7694

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 7059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7059	. . . . . 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 3181	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))
8		oveq1 7397	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
9	8	eqeq1d 2732	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦) = 𝑦 ↔ ((𝐹‘𝑤)𝑅𝑦) = 𝑦))
10		eqeq1 2734	. . . . . . . 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 7398	. . . . . . . . 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 2746	. . . . . . . 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 3602	. . . . . 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 3155	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = (𝐺‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) = 0 ))
20		ovexd 7425	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
21	20	ralrimiva 3126	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
22		mpteqb 6990	. . . 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 3126	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤) ∈ 𝑆)
25		mpteqb 6990	. . . 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 6932	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	3	feqmptd 6932	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	28, 2, 4, 29, 30	offval2 7676	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	31, 30	eqeq12d 2746	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))))
33		fconstmpt 5703	. . . 4 ⊢ (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 )
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × { 0 }) = (𝑤 ∈ 𝐴 ↦ 0 ))
35	29, 34	eqeq12d 2746	. 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 3045  Vcvv 3450  {csn 4592   ↦ cmpt 5191   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656

This theorem is referenced by:  psdmvr  22063