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Theorem caofid0l 7441
 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 6557 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
65ffnd 6504 . 2 (𝜑𝐹 Fn 𝐴)
7 fvconst2g 6961 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
82, 7sylan 583 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
9 eqidd 2759 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
10 caofid0l.5 . . . 4 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)
1110ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥)
125ffvelrnda 6848 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
13 oveq2 7164 . . . . 5 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
14 id 22 . . . . 5 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1513, 14eqeq12d 2774 . . . 4 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤)))
1615rspccva 3542 . . 3 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1711, 12, 16syl2an2r 684 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
181, 4, 6, 6, 8, 9, 17offveq 7434 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {csn 4525   × cxp 5526   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘f cof 7409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411 This theorem is referenced by:  psr0lid  20736  psrlmod  20742  mndvlid  21108  lfladd0l  36684  mendlmod  40545
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