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Theorem caofid1 7724
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)
32ffnd 6728 . 2 (𝜑𝐹 Fn 𝐴)
4 caofid0.3 . . 3 (𝜑𝐵𝑊)
5 fnconstg 6790 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
64, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6790 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 eqidd 2729 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
11 fvconst2g 7220 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
124, 11sylan 578 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13 caofid1.5 . . . . 5 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)
1413ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶)
152ffvelcdmda 7099 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq1 7433 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
1716eqeq1d 2730 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹𝑤)𝑅𝐵) = 𝐶))
1817rspccva 3610 . . . 4 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
1914, 15, 18syl2an2r 683 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
20 fvconst2g 7220 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 578 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2771 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
231, 3, 6, 9, 10, 12, 22offveq 7715 1 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  {csn 4632   × cxp 5680   Fn wfn 6548  wf 6549  cfv 6553  (class class class)co 7426  f cof 7689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691
This theorem is referenced by:  plymul0or  26235  fta1lem  26262  lfl0sc  38586
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