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Theorem caofid2 7656
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 6735 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
65ffnd 6674 . 2 (𝜑𝐹 Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6735 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 fvconst2g 7156 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
112, 10sylan 581 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12 eqidd 2738 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
13 caofid2.5 . . . . 5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
1413ralrimiva 3144 . . . 4 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶)
155ffvelcdmda 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq2 7370 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
1716eqeq1d 2739 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹𝑤)) = 𝐶))
1817rspccva 3583 . . . 4 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
1914, 15, 18syl2an2r 684 . . 3 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
20 fvconst2g 7156 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 581 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2780 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = ((𝐴 × {𝐶})‘𝑤))
231, 4, 6, 9, 11, 12, 22offveq 7646 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  {csn 4591   × cxp 5636   Fn wfn 6496  wf 6497  cfv 6501  (class class class)co 7362  f cof 7620
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  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 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  mbfmulc2lem  25027  i1fmulc  25084  itg1mulc  25085  itg2mulc  25128  dvcmulf  25325  coe0  25633  plymul0or  25657  0prjspnrel  40994  expgrowth  42689
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