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Theorem caofid2 6893
 Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofid0.3 (𝜑𝐵𝑊)
caofid1.4 (𝜑𝐶𝑋)
caofid2.5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
Assertion
Ref Expression
caofid2 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (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 caofid2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2 (𝜑𝐴𝑉)
2 caofid0.3 . . 3 (𝜑𝐵𝑊)
3 fnconstg 6060 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
6 ffn 6012 . . 3 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝐴)
8 caofid1.4 . . 3 (𝜑𝐶𝑋)
9 fnconstg 6060 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
108, 9syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
11 fvconst2g 6432 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
122, 11sylan 488 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13 eqidd 2622 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
145ffvelrnda 6325 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
15 caofid2.5 . . . . . 6 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
1615ralrimiva 2962 . . . . 5 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶)
17 oveq2 6623 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
1817eqeq1d 2623 . . . . . 6 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹𝑤)) = 𝐶))
1918rspccva 3298 . . . . 5 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
2016, 19sylan 488 . . . 4 ((𝜑 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
2114, 20syldan 487 . . 3 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
22 fvconst2g 6432 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
238, 22sylan 488 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2421, 23eqtr4d 2658 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = ((𝐴 × {𝐶})‘𝑤))
251, 4, 7, 10, 12, 13, 24offveq 6883 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  {csn 4155   × cxp 5082   Fn wfn 5852  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ∘𝑓 cof 6860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862 This theorem is referenced by:  mbfmulc2lem  23354  i1fmulc  23410  itg1mulc  23411  itg2mulc  23454  dvcmulf  23648  coe0  23950  plymul0or  23974  expgrowth  38055
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