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Theorem caofid1 7428
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 6508 . 2 (𝜑𝐹 Fn 𝐴)
4 caofid0.3 . . 3 (𝜑𝐵𝑊)
5 fnconstg 6560 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
64, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6560 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 eqidd 2819 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
11 fvconst2g 6956 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
124, 11sylan 580 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13 caofid1.5 . . . . 5 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)
1413ralrimiva 3179 . . . 4 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶)
152ffvelrnda 6843 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq1 7152 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
1716eqeq1d 2820 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹𝑤)𝑅𝐵) = 𝐶))
1817rspccva 3619 . . . 4 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
1914, 15, 18syl2an2r 681 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
20 fvconst2g 6956 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 580 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2856 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
231, 3, 6, 9, 10, 12, 22offveq 7419 1 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {csn 4557   × cxp 5546   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  plymul0or  24797  fta1lem  24823  lfl0sc  36098
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