Theorem caofid1 7628

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 6652	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fnconstg 6713	. . 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 6713	. . 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 7133	. . 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 3139	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶)
15	2	ffvelcdmda 7017	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq1 7344	. . . . . 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 3569	. . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
19	14, 15, 18	syl2an2r 682	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
20		fvconst2g 7133	. . . 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 7619	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  {csn 4573   × cxp 5618   Fn wfn 6474  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337   ∘_f cof 7593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595

This theorem is referenced by:  plymul0or  25547  fta1lem  25573  lfl0sc  37349