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Theorem caofinvl 7729
Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofinv.3 (𝜑𝐵𝑊)
caofinv.4 (𝜑𝑁:𝑆𝑆)
caofinv.5 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
caofinvl.6 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
Assertion
Ref Expression
caofinvl (𝜑 → (𝐺f 𝑅𝐹) = (𝐴 × {𝐵}))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑣,𝐴   𝑣,𝐹,𝑥   𝑥,𝑁,𝑣   𝑣,𝑆   𝜑,𝑣
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑣)   𝑅(𝑣)   𝐺(𝑣)   𝑉(𝑥,𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem caofinvl
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . . . 4 (𝜑𝐴𝑉)
2 caofinv.5 . . . . . 6 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
3 caofinv.4 . . . . . . . 8 (𝜑𝑁:𝑆𝑆)
43adantr 480 . . . . . . 7 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
5 caofref.2 . . . . . . . 8 (𝜑𝐹:𝐴𝑆)
65ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
74, 6ffvelcdmd 7105 . . . . . 6 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
82, 7fmpt3d 7136 . . . . 5 (𝜑𝐺:𝐴𝑆)
98ffvelcdmda 7104 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
105ffvelcdmda 7104 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
11 fvex 6919 . . . . . . 7 (𝑁‘(𝐹𝑣)) ∈ V
12 eqid 2737 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
1311, 12fnmpti 6711 . . . . . 6 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴
142fneq1d 6661 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1513, 14mpbiri 258 . . . . 5 (𝜑𝐺 Fn 𝐴)
16 dffn5 6967 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1715, 16sylib 218 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
185feqmptd 6977 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
191, 9, 10, 17, 18offval2 7717 . . 3 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
202fveq1d 6908 . . . . . . 7 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
21 2fveq3 6911 . . . . . . . 8 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
22 fvex 6919 . . . . . . . 8 (𝑁‘(𝐹𝑤)) ∈ V
2321, 12, 22fvmpt 7016 . . . . . . 7 (𝑤𝐴 → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
2420, 23sylan9eq 2797 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
2524oveq1d 7446 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
26 fveq2 6906 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
27 id 22 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
2826, 27oveq12d 7449 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
2928eqeq1d 2739 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
30 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3130ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3231adantr 480 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3329, 32, 10rspcdva 3623 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
3425, 33eqtrd 2777 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
3534mpteq2dva 5242 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
3619, 35eqtrd 2777 . 2 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴𝐵))
37 fconstmpt 5747 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
3836, 37eqtr4di 2795 1 (𝜑 → (𝐺f 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {csn 4626  cmpt 5225   × cxp 5683   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  grpvlinv  22402  lflnegl  39077
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