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Theorem caofid1 6969
 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)
caofid1.5 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)
Assertion
Ref Expression
caofid1 (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = (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 caofid1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2 (𝜑𝐴𝑉)
2 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
3 ffn 6083 . . 3 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
42, 3syl 17 . 2 (𝜑𝐹 Fn 𝐴)
5 caofid0.3 . . 3 (𝜑𝐵𝑊)
6 fnconstg 6131 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
75, 6syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
8 caofid1.4 . . 3 (𝜑𝐶𝑋)
9 fnconstg 6131 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
108, 9syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
11 eqidd 2652 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
12 fvconst2g 6508 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
135, 12sylan 487 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
142ffvelrnda 6399 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
15 caofid1.5 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)
1615ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶)
17 oveq1 6697 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
1817eqeq1d 2653 . . . . . 6 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹𝑤)𝑅𝐵) = 𝐶))
1918rspccva 3339 . . . . 5 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
2016, 19sylan 487 . . . 4 ((𝜑 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
2114, 20syldan 486 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
22 fvconst2g 6508 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
238, 22sylan 487 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2421, 23eqtr4d 2688 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
251, 4, 7, 10, 11, 13, 24offveq 6960 1 (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {csn 4210   × cxp 5141   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939 This theorem is referenced by:  plymul0or  24081  fta1lem  24107  lfl0sc  34687
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