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Theorem caofinvl 5784
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 5354 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
63, 5ffvelrnd 5355 . . . . . . 7 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
7 eqid 2083 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
86, 7fmptd 5374 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆)
9 caofinv.5 . . . . . . 7 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
109feq1d 5085 . . . . . 6 (𝜑 → (𝐺:𝐴𝑆 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆))
118, 10mpbird 165 . . . . 5 (𝜑𝐺:𝐴𝑆)
1211ffvelrnda 5354 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
134ffvelrnda 5354 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
146ralrimiva 2439 . . . . . . 7 (𝜑 → ∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆)
157fnmpt 5076 . . . . . . 7 (∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
1614, 15syl 14 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
179fneq1d 5040 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1816, 17mpbird 165 . . . . 5 (𝜑𝐺 Fn 𝐴)
19 dffn5im 5271 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
2018, 19syl 14 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
214feqmptd 5278 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
221, 12, 13, 20, 21offval2 5777 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
239fveq1d 5231 . . . . . . . 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 5355 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝑁‘(𝐹𝑤)) ∈ 𝑆)
28 fveq2 5229 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
2928fveq2d 5233 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
3029, 7fvmptg 5300 . . . . . . . 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 2115 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
3332oveq1d 5578 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
34 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3534ralrimiva 2439 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3635adantr 270 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
37 fveq2 5229 . . . . . . . . 9 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
38 id 19 . . . . . . . . 9 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
3937, 38oveq12d 5581 . . . . . . . 8 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
4039eqeq1d 2091 . . . . . . 7 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
4140rspcva 2708 . . . . . 6 (((𝐹𝑤) ∈ 𝑆 ∧ ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
4213, 36, 41syl2anc 403 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
4333, 42eqtrd 2115 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
4443mpteq2dva 3888 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
4522, 44eqtrd 2115 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴𝐵))
46 fconstmpt 4433 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
4745, 46syl6eqr 2133 1 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wral 2353  {csn 3416  cmpt 3859   × cxp 4389   Fn wfn 4947  wf 4948  cfv 4952  (class class class)co 5563  𝑓 cof 5761
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-of 5763
This theorem is referenced by: (None)
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