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Theorem caofid0l 6823
 Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofid0.3 (𝜑𝐵𝑊)
caofid0l.5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)
Assertion
Ref Expression
caofid0l (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem caofid0l
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2 (𝜑𝐴𝑉)
2 caofid0.3 . . 3 (𝜑𝐵𝑊)
3 fnconstg 6006 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
6 ffn 5958 . . 3 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝐴)
8 fvconst2g 6372 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
92, 8sylan 487 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
10 eqidd 2611 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
115ffvelrnda 6267 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
12 caofid0l.5 . . . . 5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)
1312ralrimiva 2949 . . . 4 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥)
14 oveq2 6557 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
15 id 22 . . . . . 6 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1614, 15eqeq12d 2625 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤)))
1716rspccva 3281 . . . 4 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1813, 17sylan 487 . . 3 ((𝜑 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1911, 18syldan 486 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
201, 4, 7, 7, 9, 10, 19offveq 6816 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125   × cxp 5036   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by:  psr0lid  19216  psrlmod  19222  mndvlid  20018  lfladd0l  33379  mendlmod  36782
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