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Theorem caofid1 7438
 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 6500 . 2 (𝜑𝐹 Fn 𝐴)
4 caofid0.3 . . 3 (𝜑𝐵𝑊)
5 fnconstg 6553 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
64, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7 caofid1.4 . . 3 (𝜑𝐶𝑋)
8 fnconstg 6553 . . 3 (𝐶𝑋 → (𝐴 × {𝐶}) Fn 𝐴)
97, 8syl 17 . 2 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 eqidd 2760 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
11 fvconst2g 6956 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
124, 11sylan 584 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
13 caofid1.5 . . . . 5 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)
1413ralrimiva 3114 . . . 4 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶)
152ffvelrnda 6843 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
16 oveq1 7158 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
1716eqeq1d 2761 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝐶 ↔ ((𝐹𝑤)𝑅𝐵) = 𝐶))
1817rspccva 3541 . . . 4 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝐶 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
1914, 15, 18syl2an2r 685 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = 𝐶)
20 fvconst2g 6956 . . . 4 ((𝐶𝑋𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
217, 20sylan 584 . . 3 ((𝜑𝑤𝐴) → ((𝐴 × {𝐶})‘𝑤) = 𝐶)
2219, 21eqtr4d 2797 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = ((𝐴 × {𝐶})‘𝑤))
231, 3, 6, 9, 10, 12, 22offveq 7429 1 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {csn 4523   × cxp 5523   Fn wfn 6331  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ∘f cof 7404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7406 This theorem is referenced by:  plymul0or  24969  fta1lem  24995  lfl0sc  36651
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