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Theorem caofinvl 6105
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 276 . . . . . . . 8 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
4 caofref.2 . . . . . . . . 9 (𝜑𝐹:𝐴𝑆)
54ffvelcdmda 5652 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
63, 5ffvelcdmd 5653 . . . . . . 7 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
7 eqid 2177 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
86, 7fmptd 5671 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆)
9 caofinv.5 . . . . . . 7 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
109feq1d 5353 . . . . . 6 (𝜑 → (𝐺:𝐴𝑆 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆))
118, 10mpbird 167 . . . . 5 (𝜑𝐺:𝐴𝑆)
1211ffvelcdmda 5652 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
134ffvelcdmda 5652 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
146ralrimiva 2550 . . . . . . 7 (𝜑 → ∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆)
157fnmpt 5343 . . . . . . 7 (∀𝑣𝐴 (𝑁‘(𝐹𝑣)) ∈ 𝑆 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
1614, 15syl 14 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴)
179fneq1d 5307 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1816, 17mpbird 167 . . . . 5 (𝜑𝐺 Fn 𝐴)
19 dffn5im 5562 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
2018, 19syl 14 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
214feqmptd 5570 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
221, 12, 13, 20, 21offval2 6098 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
239fveq1d 5518 . . . . . . . 8 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
2423adantr 276 . . . . . . 7 ((𝜑𝑤𝐴) → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
25 simpr 110 . . . . . . . 8 ((𝜑𝑤𝐴) → 𝑤𝐴)
262adantr 276 . . . . . . . . 9 ((𝜑𝑤𝐴) → 𝑁:𝑆𝑆)
2726, 13ffvelcdmd 5653 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝑁‘(𝐹𝑤)) ∈ 𝑆)
28 fveq2 5516 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
2928fveq2d 5520 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
3029, 7fvmptg 5593 . . . . . . . 8 ((𝑤𝐴 ∧ (𝑁‘(𝐹𝑤)) ∈ 𝑆) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3125, 27, 30syl2anc 411 . . . . . . 7 ((𝜑𝑤𝐴) → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
3224, 31eqtrd 2210 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
3332oveq1d 5890 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
34 fveq2 5516 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
35 id 19 . . . . . . . 8 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
3634, 35oveq12d 5893 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
3736eqeq1d 2186 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
38 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3938ralrimiva 2550 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4039adantr 276 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
4137, 40, 13rspcdva 2847 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
4233, 41eqtrd 2210 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
4342mpteq2dva 4094 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
4422, 43eqtrd 2210 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴𝐵))
45 fconstmpt 4674 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
4644, 45eqtr4di 2228 1 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  {csn 3593  cmpt 4065   × cxp 4625   Fn wfn 5212  wf 5213  cfv 5217  (class class class)co 5875  𝑓 cof 6081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083
This theorem is referenced by: (None)
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