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Theorem caofid2 7567
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 6662 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
65ffnd 6601 . 2 (𝜑𝐹 Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6662 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 fvconst2g 7077 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
112, 10sylan 580 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12 eqidd 2739 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
13 caofid2.5 . . . . 5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
1413ralrimiva 3103 . . . 4 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶)
155ffvelrnda 6961 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq2 7283 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
1716eqeq1d 2740 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹𝑤)) = 𝐶))
1817rspccva 3560 . . . 4 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
1914, 15, 18syl2an2r 682 . . 3 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
20 fvconst2g 7077 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 580 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2781 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = ((𝐴 × {𝐶})‘𝑤))
231, 4, 6, 9, 11, 12, 22offveq 7557 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  {csn 4561   × cxp 5587   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533
This theorem is referenced by:  mbfmulc2lem  24811  i1fmulc  24868  itg1mulc  24869  itg2mulc  24912  dvcmulf  25109  coe0  25417  plymul0or  25441  0prjspnrel  40464  expgrowth  41953
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