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Theorem invffval 17701
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 6881 . . . . . 6 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
4 invfval.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
53, 4eqtr4di 2782 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
6 fveq2 6881 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Sectβ€˜π‘) = (Sectβ€˜πΆ))
7 invfval.s . . . . . . . 8 𝑆 = (Sectβ€˜πΆ)
86, 7eqtr4di 2782 . . . . . . 7 (𝑐 = 𝐢 β†’ (Sectβ€˜π‘) = 𝑆)
98oveqd 7418 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘₯(Sectβ€˜π‘)𝑦) = (π‘₯𝑆𝑦))
108oveqd 7418 . . . . . . 7 (𝑐 = 𝐢 β†’ (𝑦(Sectβ€˜π‘)π‘₯) = (𝑦𝑆π‘₯))
1110cnveqd 5865 . . . . . 6 (𝑐 = 𝐢 β†’ β—‘(𝑦(Sectβ€˜π‘)π‘₯) = β—‘(𝑦𝑆π‘₯))
129, 11ineq12d 4205 . . . . 5 (𝑐 = 𝐢 β†’ ((π‘₯(Sectβ€˜π‘)𝑦) ∩ β—‘(𝑦(Sectβ€˜π‘)π‘₯)) = ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯)))
135, 5, 12mpoeq123dv 7476 . . . 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 17691 . . . 4 Inv = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘), 𝑦 ∈ (Baseβ€˜π‘) ↦ ((π‘₯(Sectβ€˜π‘)𝑦) ∩ β—‘(𝑦(Sectβ€˜π‘)π‘₯))))
154fvexi 6895 . . . . 5 𝐡 ∈ V
1615, 15mpoex 8059 . . . 4 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ((π‘₯𝑆𝑦) ∩ β—‘(𝑦𝑆π‘₯))) ∈ V
1713, 14, 16fvmpt 6988 . . 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 2776 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 1533   ∈ wcel 2098   ∩ cin 3939  β—‘ccnv 5665  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  Basecbs 17140  Catccat 17604  Sectcsect 17687  Invcinv 17688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-inv 17691
This theorem is referenced by:  invfval  17702  isoval  17708
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