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Theorem caofinvl 5859
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.4 . . . . . . . . 9 (𝜑𝑁:𝑆𝑆)
32adantr 270 . . . . . . . 8 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
4 caofref.2 . . . . . . . . 9 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 5418 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
63, 5ffvelrnd 5419 . . . . . . 7 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
7 eqid 2088 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
86, 7fmptd 5436 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆)
9 caofinv.5 . . . . . . 7 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
109feq1d 5135 . . . . . 6 (𝜑 → (𝐺:𝐴𝑆 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆))
118, 10mpbird 165 . . . . 5 (𝜑𝐺:𝐴𝑆)
1211ffvelrnda 5418 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
134ffvelrnda 5418 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
146ralrimiva 2446 . . . . . . 7 (𝜑 → ∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆)
157fnmpt 5126 . . . . . . 7 (∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
1614, 15syl 14 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
179fneq1d 5090 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1816, 17mpbird 165 . . . . 5 (𝜑𝐺 Fn 𝐴)
19 dffn5im 5334 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
2018, 19syl 14 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
214feqmptd 5341 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
221, 12, 13, 20, 21offval2 5852 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
239fveq1d 5291 . . . . . . . 8 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
2423adantr 270 . . . . . . 7 ((𝜑𝑤𝐴) → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
25 simpr 108 . . . . . . . 8 ((𝜑𝑤𝐴) → 𝑤𝐴)
262adantr 270 . . . . . . . . 9 ((𝜑𝑤𝐴) → 𝑁:𝑆𝑆)
2726, 13ffvelrnd 5419 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝑁‘(𝐹𝑤)) ∈ 𝑆)
28 fveq2 5289 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
2928fveq2d 5293 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
3029, 7fvmptg 5364 . . . . . . . 8 ((𝑤𝐴 ∧ (𝑁‘(𝐹𝑤)) ∈ 𝑆) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3125, 27, 30syl2anc 403 . . . . . . 7 ((𝜑𝑤𝐴) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3224, 31eqtrd 2120 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
3332oveq1d 5649 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
34 fveq2 5289 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
35 id 19 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
3634, 35oveq12d 5652 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
3736eqeq1d 2096 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
38 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3938ralrimiva 2446 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4039adantr 270 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4137, 40, 13rspcdva 2727 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
4233, 41eqtrd 2120 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
4342mpteq2dva 3920 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
4422, 43eqtrd 2120 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴𝐵))
45 fconstmpt 4473 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
4644, 45syl6eqr 2138 1 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wral 2359  {csn 3441  cmpt 3891   × cxp 4426   Fn wfn 4997  wf 4998  cfv 5002  (class class class)co 5634  𝑓 cof 5836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-of 5838
This theorem is referenced by: (None)
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