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Theorem caofid0l 6879
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 6052 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
6 ffn 6004 . . 3 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
75, 6syl 17 . 2 (𝜑𝐹 Fn 𝐴)
8 fvconst2g 6422 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
92, 8sylan 488 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
10 eqidd 2627 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
115ffvelrnda 6316 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
12 caofid0l.5 . . . . 5 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)
1312ralrimiva 2965 . . . 4 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥)
14 oveq2 6613 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
15 id 22 . . . . . 6 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1614, 15eqeq12d 2641 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤)))
1716rspccva 3299 . . . 4 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1813, 17sylan 488 . . 3 ((𝜑 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1911, 18syldan 487 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
201, 4, 7, 7, 9, 10, 19offveq 6872 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  {csn 4153   × cxp 5077   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  𝑓 cof 6849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851
This theorem is referenced by:  psr0lid  19309  psrlmod  19315  mndvlid  20113  lfladd0l  33827  mendlmod  37230
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