Theorem caofid1 7733

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)
caofid1.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶)

Assertion

Ref	Expression
caofid1	⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (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 caofid1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		caofref.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
3	2	ffnd 6736	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fnconstg 6795	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7		caofid1.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
8		fnconstg 6795	. . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10		eqidd 2737	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
11		fvconst2g 7223	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12	4, 11	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13		caofid1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶)
14	13	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶)
15	2	ffvelcdmda 7103	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq1 7439	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵))
17	16	eqeq1d 2738	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹‘𝑤)𝑅𝐵) = 𝐶))
18	17	rspccva 3620	. . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
19	14, 15, 18	syl2an2r 685	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
20		fvconst2g 7223	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
21	7, 20	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
22	19, 21	eqtr4d 2779	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
23	1, 3, 6, 9, 10, 12, 22	offveq 7724	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  {csn 4625   × cxp 5682   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∘_f cof 7696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698

This theorem is referenced by:  plymul0or  26323  fta1lem  26350  lfl0sc  39084