Theorem caofid2 6218

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,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶

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 5499	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5		caofref.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6	5	ffnd 5450	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		caofid1.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
8		fnconstg 5499	. . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10		fvconst2g 5826	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
11	2, 10	sylan 283	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12		eqidd 2210	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
13		caofid2.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶)
14	13	ralrimiva 2583	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶)
15	5	ffvelcdmda 5743	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq2 5982	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹‘𝑤)))
17	16	eqeq1d 2218	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹‘𝑤)) = 𝐶))
18	17	rspccva 2886	. . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶)
19	14, 15, 18	syl2an2r 597	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = 𝐶)
20		fvconst2g 5826	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
21	7, 20	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
22	19, 21	eqtr4d 2245	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐵𝑅(𝐹‘𝑤)) = ((𝐴 × {𝐶})‘𝑤))
23	1, 4, 6, 9, 11, 12, 22	offveq 6209	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∀wral 2488  {csn 3646   × cxp 4694   Fn wfn 5289  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974   ∘_𝑓 cof 6186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188

This theorem is referenced by: (None)