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Theorem caofinvl 6878
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 481 . . . . . . . 8 ((𝜑𝑣𝐴) → 𝑁:𝑆𝑆)
4 caofref.2 . . . . . . . . 9 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6316 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹𝑣) ∈ 𝑆)
63, 5ffvelrnd 6317 . . . . . . 7 ((𝜑𝑣𝐴) → (𝑁‘(𝐹𝑣)) ∈ 𝑆)
7 eqid 2626 . . . . . . 7 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))
86, 7fmptd 6341 . . . . . 6 (𝜑 → (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆)
9 caofinv.5 . . . . . . 7 (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))
109feq1d 5989 . . . . . 6 (𝜑 → (𝐺:𝐴𝑆 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))):𝐴𝑆))
118, 10mpbird 247 . . . . 5 (𝜑𝐺:𝐴𝑆)
1211ffvelrnda 6316 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
134ffvelrnda 6316 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
14 fvex 6160 . . . . . . 7 (𝑁‘(𝐹𝑣)) ∈ V
1514, 7fnmpti 5981 . . . . . 6 (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴
169fneq1d 5941 . . . . . 6 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))) Fn 𝐴))
1715, 16mpbiri 248 . . . . 5 (𝜑𝐺 Fn 𝐴)
18 dffn5 6199 . . . . 5 (𝐺 Fn 𝐴𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1917, 18sylib 208 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
204feqmptd 6207 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
211, 12, 13, 19, 20offval2 6868 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
229fveq1d 6152 . . . . . . 7 (𝜑 → (𝐺𝑤) = ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤))
23 fveq2 6150 . . . . . . . . 9 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
2423fveq2d 6154 . . . . . . . 8 (𝑣 = 𝑤 → (𝑁‘(𝐹𝑣)) = (𝑁‘(𝐹𝑤)))
25 fvex 6160 . . . . . . . 8 (𝑁‘(𝐹𝑤)) ∈ V
2624, 7, 25fvmpt 6240 . . . . . . 7 (𝑤𝐴 → ((𝑣𝐴 ↦ (𝑁‘(𝐹𝑣)))‘𝑤) = (𝑁‘(𝐹𝑤)))
2722, 26sylan9eq 2680 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝑁‘(𝐹𝑤)))
2827oveq1d 6620 . . . . 5 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
29 caofinvl.6 . . . . . . . 8 ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)
3029ralrimiva 2965 . . . . . . 7 (𝜑 → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
3130adantr 481 . . . . . 6 ((𝜑𝑤𝐴) → ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵)
32 fveq2 6150 . . . . . . . . 9 (𝑥 = (𝐹𝑤) → (𝑁𝑥) = (𝑁‘(𝐹𝑤)))
33 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
3432, 33oveq12d 6623 . . . . . . . 8 (𝑥 = (𝐹𝑤) → ((𝑁𝑥)𝑅𝑥) = ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)))
3534eqeq1d 2628 . . . . . . 7 (𝑥 = (𝐹𝑤) → (((𝑁𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵))
3635rspcva 3298 . . . . . 6 (((𝐹𝑤) ∈ 𝑆 ∧ ∀𝑥𝑆 ((𝑁𝑥)𝑅𝑥) = 𝐵) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
3713, 31, 36syl2anc 692 . . . . 5 ((𝜑𝑤𝐴) → ((𝑁‘(𝐹𝑤))𝑅(𝐹𝑤)) = 𝐵)
3828, 37eqtrd 2660 . . . 4 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐹𝑤)) = 𝐵)
3938mpteq2dva 4709 . . 3 (𝜑 → (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))) = (𝑤𝐴𝐵))
4021, 39eqtrd 2660 . 2 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝑤𝐴𝐵))
41 fconstmpt 5128 . 2 (𝐴 × {𝐵}) = (𝑤𝐴𝐵)
4240, 41syl6eqr 2678 1 (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  {csn 4153  cmpt 4678   × cxp 5077   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  𝑓 cof 6849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851
This theorem is referenced by:  grpvlinv  20115  lflnegl  33829
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