Theorem caofid2 7545

Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofid0.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)
caofid1.4	⊢ (𝜑 → 𝐶 ∈ 𝑋)
caofid2.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶)

Assertion

Ref	Expression
caofid2	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶}))

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

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

Proof of Theorem caofid2

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 6646	. . 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 6585	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		caofid1.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
8		fnconstg 6646	. . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10		fvconst2g 7059	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
11	2, 10	sylan 579	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12		eqidd 2739	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
13		caofid2.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶)
14	13	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶)
15	5	ffvelrnda 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq2 7263	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤)))
17	16	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹‘𝑤)) = 𝐶))
18	17	rspccva 3551	. . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶)
19	14, 15, 18	syl2an2r 681	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶)
20		fvconst2g 7059	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
21	7, 20	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
22	19, 21	eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = ((𝐴 × {𝐶})‘𝑤))
23	1, 4, 6, 9, 11, 12, 22	offveq 7535	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {csn 4558   × cxp 5578   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511

This theorem is referenced by:  mbfmulc2lem  24716  i1fmulc  24773  itg1mulc  24774  itg2mulc  24817  dvcmulf  25014  coe0  25322  plymul0or  25346  0prjspnrel  40385  expgrowth  41842