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Theorem caofid2 7443
 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 6570 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
65ffnd 6518 . 2 (𝜑𝐹 Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6570 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 fvconst2g 6967 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
112, 10sylan 582 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
12 eqidd 2825 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
13 caofid2.5 . . . . 5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
1413ralrimiva 3185 . . . 4 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶)
155ffvelrnda 6854 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq2 7167 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
1716eqeq1d 2826 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹𝑤)) = 𝐶))
1817rspccva 3625 . . . 4 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
1914, 15, 18syl2an2r 683 . . 3 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
20 fvconst2g 6967 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 582 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2862 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = ((𝐴 × {𝐶})‘𝑤))
231, 4, 6, 9, 11, 12, 22offveq 7433 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {csn 4570   × cxp 5556   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ∘f cof 7410 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412 This theorem is referenced by:  mbfmulc2lem  24251  i1fmulc  24307  itg1mulc  24308  itg2mulc  24351  dvcmulf  24545  coe0  24849  plymul0or  24873  0prjspnrel  39275  expgrowth  40673
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