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Theorem caofinvl 7429
 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 484 . . . . . . 7 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
5 caofref.2 . . . . . . . 8 (𝜑𝐹:𝐴𝑆)
65ffvelrnda 6838 . . . . . . 7 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
74, 6ffvelrnd 6839 . . . . . 6 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
82, 7fmpt3d 6867 . . . . 5 (𝜑𝐺:𝐴𝑆)
98ffvelrnda 6838 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
105ffvelrnda 6838 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
11 fvex 6668 . . . . . . 7 (𝑁‘(𝐹𝑣)) ∈ V
12 eqid 2798 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
1311, 12fnmpti 6471 . . . . . 6 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴
142fneq1d 6424 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1513, 14mpbiri 261 . . . . 5 (𝜑𝐺 Fn 𝐴)
16 dffn5 6709 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1715, 16sylib 221 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
185feqmptd 6718 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
191, 9, 10, 17, 18offval2 7419 . . 3 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
202fveq1d 6657 . . . . . . 7 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
21 2fveq3 6660 . . . . . . . 8 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
22 fvex 6668 . . . . . . . 8 (𝑁‘(𝐹𝑤)) ∈ V
2321, 12, 22fvmpt 6755 . . . . . . 7 (𝑤𝐴 → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
2420, 23sylan9eq 2853 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
2524oveq1d 7160 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
26 fveq2 6655 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
27 id 22 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
2826, 27oveq12d 7163 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
2928eqeq1d 2800 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
30 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3130ralrimiva 3149 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3231adantr 484 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3329, 32, 10rspcdva 3574 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
3425, 33eqtrd 2833 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
3534mpteq2dva 5129 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
3619, 35eqtrd 2833 . 2 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴𝐵))
37 fconstmpt 5582 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
3836, 37eqtr4di 2851 1 (𝜑 → (𝐺f 𝑅𝐹) = (𝐴 × {𝐵}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {csn 4528   ↦ cmpt 5114   × cxp 5521   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ∘f cof 7398 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 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400 This theorem is referenced by:  grpvlinv  21043  lflnegl  36523
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