Theorem caofid1 6269

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,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶

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 5485	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fnconstg 5537	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7		caofid1.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
8		fnconstg 5537	. . 3 ⊢ (𝐶 ∈ 𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10		eqidd 2231	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
11		fvconst2g 5871	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12	4, 11	sylan 283	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13		caofid1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶)
14	13	ralrimiva 2604	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶)
15	2	ffvelcdmda 5785	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
16		oveq1 6030	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵))
17	16	eqeq1d 2239	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹‘𝑤)𝑅𝐵) = 𝐶))
18	17	rspccva 2908	. . . 4 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
19	14, 15, 18	syl2an2r 599	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = 𝐶)
20		fvconst2g 5871	. . . 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 2266	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
23	1, 3, 6, 9, 10, 12, 22	offveq 6261	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ∀wral 2509  {csn 3670   × cxp 4725   Fn wfn 5323  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023   ∘_𝑓 cof 6238

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-of 6240

This theorem is referenced by: (None)