Theorem caofid1 7688

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 6689	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fnconstg 6748	. . 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 6748	. . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10		eqidd 2730	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
11		fvconst2g 7176	. . 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 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶)
15	2	ffvelcdmda 7056	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq1 7394	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵))
17	16	eqeq1d 2731	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹‘𝑤)𝑅𝐵) = 𝐶))
18	17	rspccva 3587	. . . 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 7176	. . . 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 2767	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
23	1, 3, 6, 9, 10, 12, 22	offveq 7679	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {csn 4589   × cxp 5636   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by:  plymul0or  26188  fta1lem  26215  lfl0sc  39075