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Theorem caofinvl 7721
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 479 . . . . . . 7 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
5 caofref.2 . . . . . . . 8 (𝜑𝐹:𝐴𝑆)
65ffvelcdmda 7098 . . . . . . 7 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
74, 6ffvelcdmd 7099 . . . . . 6 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
82, 7fmpt3d 7130 . . . . 5 (𝜑𝐺:𝐴𝑆)
98ffvelcdmda 7098 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
105ffvelcdmda 7098 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
11 fvex 6914 . . . . . . 7 (𝑁‘(𝐹𝑣)) ∈ V
12 eqid 2726 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
1311, 12fnmpti 6704 . . . . . 6 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴
142fneq1d 6653 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1513, 14mpbiri 257 . . . . 5 (𝜑𝐺 Fn 𝐴)
16 dffn5 6961 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1715, 16sylib 217 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
185feqmptd 6971 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
191, 9, 10, 17, 18offval2 7710 . . 3 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
202fveq1d 6903 . . . . . . 7 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
21 2fveq3 6906 . . . . . . . 8 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
22 fvex 6914 . . . . . . . 8 (𝑁‘(𝐹𝑤)) ∈ V
2321, 12, 22fvmpt 7009 . . . . . . 7 (𝑤𝐴 → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
2420, 23sylan9eq 2786 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
2524oveq1d 7439 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
26 fveq2 6901 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
27 id 22 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
2826, 27oveq12d 7442 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
2928eqeq1d 2728 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
30 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3130ralrimiva 3136 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3231adantr 479 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3329, 32, 10rspcdva 3609 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
3425, 33eqtrd 2766 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
3534mpteq2dva 5253 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
3619, 35eqtrd 2766 . 2 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴𝐵))
37 fconstmpt 5744 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
3836, 37eqtr4di 2784 1 (𝜑 → (𝐺f 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  {csn 4633  cmpt 5236   × cxp 5680   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  f cof 7688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690
This theorem is referenced by:  grpvlinv  22389  lflnegl  38774
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