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Theorem caofid2 7075
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 6233 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
6 ffn 6185 . . 3 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝐴)
8 caofid1.4 . . 3 (𝜑𝐶𝑋)
9 fnconstg 6233 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
108, 9syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
11 fvconst2g 6611 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
122, 11sylan 569 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13 eqidd 2772 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
145ffvelrnda 6502 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
15 caofid2.5 . . . . . 6 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)
1615ralrimiva 3115 . . . . 5 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶)
17 oveq2 6801 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
1817eqeq1d 2773 . . . . . 6 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝐶 ↔ (𝐵𝑅(𝐹𝑤)) = 𝐶))
1918rspccva 3459 . . . . 5 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
2016, 19sylan 569 . . . 4 ((𝜑 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
2114, 20syldan 579 . . 3 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = 𝐶)
22 fvconst2g 6611 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
238, 22sylan 569 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2421, 23eqtr4d 2808 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = ((𝐴 × {𝐶})‘𝑤))
251, 4, 7, 10, 12, 13, 24offveq 7065 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {csn 4316   × cxp 5247   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  𝑓 cof 7042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044
This theorem is referenced by:  mbfmulc2lem  23634  i1fmulc  23690  itg1mulc  23691  itg2mulc  23734  dvcmulf  23928  coe0  24232  plymul0or  24256  expgrowth  39060
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