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Theorem caofinvl 6016
 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 274 . . . . . . . 8 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
4 caofref.2 . . . . . . . . 9 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 5567 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
63, 5ffvelrnd 5568 . . . . . . 7 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
7 eqid 2141 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
86, 7fmptd 5586 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆)
9 caofinv.5 . . . . . . 7 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
109feq1d 5271 . . . . . 6 (𝜑 → (𝐺:𝐴𝑆 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆))
118, 10mpbird 166 . . . . 5 (𝜑𝐺:𝐴𝑆)
1211ffvelrnda 5567 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
134ffvelrnda 5567 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
146ralrimiva 2510 . . . . . . 7 (𝜑 → ∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆)
157fnmpt 5261 . . . . . . 7 (∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
1614, 15syl 14 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
179fneq1d 5225 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1816, 17mpbird 166 . . . . 5 (𝜑𝐺 Fn 𝐴)
19 dffn5im 5479 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
2018, 19syl 14 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
214feqmptd 5486 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
221, 12, 13, 20, 21offval2 6009 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
239fveq1d 5435 . . . . . . . 8 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
2423adantr 274 . . . . . . 7 ((𝜑𝑤𝐴) → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
25 simpr 109 . . . . . . . 8 ((𝜑𝑤𝐴) → 𝑤𝐴)
262adantr 274 . . . . . . . . 9 ((𝜑𝑤𝐴) → 𝑁:𝑆𝑆)
2726, 13ffvelrnd 5568 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝑁‘(𝐹𝑤)) ∈ 𝑆)
28 fveq2 5433 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
2928fveq2d 5437 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
3029, 7fvmptg 5509 . . . . . . . 8 ((𝑤𝐴 ∧ (𝑁‘(𝐹𝑤)) ∈ 𝑆) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3125, 27, 30syl2anc 409 . . . . . . 7 ((𝜑𝑤𝐴) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3224, 31eqtrd 2174 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
3332oveq1d 5801 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
34 fveq2 5433 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
35 id 19 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
3634, 35oveq12d 5804 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
3736eqeq1d 2150 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
38 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3938ralrimiva 2510 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4039adantr 274 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4137, 40, 13rspcdva 2800 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
4233, 41eqtrd 2174 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
4342mpteq2dva 4028 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
4422, 43eqtrd 2174 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴𝐵))
45 fconstmpt 4598 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
4644, 45eqtr4di 2192 1 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∀wral 2418  {csn 3534   ↦ cmpt 3999   × cxp 4549   Fn wfn 5130  ⟶wf 5131  ‘cfv 5135  (class class class)co 5786   ∘𝑓 cof 5992 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-coll 4053  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-setind 4463 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-reu 2425  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-of 5994 This theorem is referenced by: (None)
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