Theorem caofid0l 7441

Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofid0.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)
caofid0l.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥)

Assertion

Ref	Expression
caofid0l	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem caofid0l

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		fnconstg 6557	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5		caofref.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6	5	ffnd 6504	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		fvconst2g 6961	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
8	2, 7	sylan 583	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
9		eqidd 2759	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
10		caofid0l.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥)
11	10	ralrimiva 3113	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥)
12	5	ffvelrnda 6848	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
13		oveq2 7164	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤)))
14		id 22	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
15	13, 14	eqeq12d 2774	. . . 4 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤)))
16	15	rspccva 3542	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤))
17	11, 12, 16	syl2an2r 684	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = (𝐹‘𝑤))
18	1, 4, 6, 6, 8, 9, 17	offveq 7434	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {csn 4525   × cxp 5526   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘_f cof 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411

This theorem is referenced by:  psr0lid  20736  psrlmod  20742  mndvlid  21108  lfladd0l  36684  mendlmod  40545