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Theorem invffval 17601
Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐡 = (Baseβ€˜πΆ)
invfval.n 𝑁 = (Invβ€˜πΆ)
invfval.c (πœ‘ β†’ 𝐢 ∈ Cat)
invfval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invfval.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
invfval.s 𝑆 = (Sectβ€˜πΆ)
Assertion
Ref Expression
invffval (πœ‘ β†’ 𝑁 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))))
Distinct variable groups:   π‘₯,𝑦,𝐡   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦   π‘₯,𝐢,𝑦   π‘₯,𝑆,𝑦
Allowed substitution hints:   𝑁(π‘₯,𝑦)

Proof of Theorem invffval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 invfval.n . 2 𝑁 = (Invβ€˜πΆ)
2 invfval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
3 fveq2 6839 . . . . . 6 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
4 invfval.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
53, 4eqtr4di 2795 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
6 fveq2 6839 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Sectβ€˜π‘) = (Sectβ€˜πΆ))
7 invfval.s . . . . . . . 8 𝑆 = (Sectβ€˜πΆ)
86, 7eqtr4di 2795 . . . . . . 7 (𝑐 = 𝐢 β†’ (Sectβ€˜π‘) = 𝑆)
98oveqd 7368 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘₯(Sectβ€˜π‘)𝑦) = (π‘₯𝑆𝑦))
108oveqd 7368 . . . . . . 7 (𝑐 = 𝐢 β†’ (𝑦(Sectβ€˜π‘)π‘₯) = (𝑦𝑆π‘₯))
1110cnveqd 5829 . . . . . 6 (𝑐 = 𝐢 β†’ β—‘(𝑦(Sectβ€˜π‘)π‘₯) = β—‘(𝑦𝑆π‘₯))
129, 11ineq12d 4171 . . . . 5 (𝑐 = 𝐢 β†’ ((π‘₯(Sectβ€˜π‘)𝑦) ∩ β—‘(𝑦(Sectβ€˜π‘)π‘₯)) = ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯)))
135, 5, 12mpoeq123dv 7426 . . . 4 (𝑐 = 𝐢 β†’ (π‘₯ ∈ (Baseβ€˜π‘), 𝑦 ∈ (Baseβ€˜π‘) ↦ ((π‘₯(Sectβ€˜π‘)𝑦) ∩ β—‘(𝑦(Sectβ€˜π‘)π‘₯))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))))
14 df-inv 17591 . . . 4 Inv = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘), 𝑦 ∈ (Baseβ€˜π‘) ↦ ((π‘₯(Sectβ€˜π‘)𝑦) ∩ β—‘(𝑦(Sectβ€˜π‘)π‘₯))))
154fvexi 6853 . . . . 5 𝐡 ∈ V
1615, 15mpoex 8004 . . . 4 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))) ∈ V
1713, 14, 16fvmpt 6945 . . 3 (𝐢 ∈ Cat β†’ (Invβ€˜πΆ) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))))
182, 17syl 17 . 2 (πœ‘ β†’ (Invβ€˜πΆ) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))))
191, 18eqtrid 2789 1 (πœ‘ β†’ 𝑁 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106   ∩ cin 3907  β—‘ccnv 5630  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  Basecbs 17043  Catccat 17504  Sectcsect 17587  Invcinv 17588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-inv 17591
This theorem is referenced by:  invfval  17602  isoval  17608
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