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Theorem caofid0l 7704
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 6779 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
65ffnd 6718 . 2 (𝜑𝐹 Fn 𝐴)
7 fvconst2g 7205 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
82, 7sylan 579 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
9 eqidd 2732 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
10 caofid0l.5 . . . 4 ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)
1110ralrimiva 3145 . . 3 (𝜑 → ∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥)
125ffvelcdmda 7086 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
13 oveq2 7420 . . . . 5 (𝑥 = (𝐹𝑤) → (𝐵𝑅𝑥) = (𝐵𝑅(𝐹𝑤)))
14 id 22 . . . . 5 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1513, 14eqeq12d 2747 . . . 4 (𝑥 = (𝐹𝑤) → ((𝐵𝑅𝑥) = 𝑥 ↔ (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤)))
1615rspccva 3611 . . 3 ((∀𝑥𝑆 (𝐵𝑅𝑥) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
1711, 12, 16syl2an2r 682 . 2 ((𝜑𝑤𝐴) → (𝐵𝑅(𝐹𝑤)) = (𝐹𝑤))
181, 4, 6, 6, 8, 9, 17offveq 7697 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  {csn 4628   × cxp 5674   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  f cof 7671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673
This theorem is referenced by:  psr0lid  21734  psrlmod  21741  mndvlid  22116  lfladd0l  38248  mendlmod  42238
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