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Theorem invinv 17112
 Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
isoval.n 𝐼 = (Iso‘𝐶)
invinv.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
invinv (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)

Proof of Theorem invinv
StepHypRef Expression
1 invfval.b . . . 4 𝐵 = (Base‘𝐶)
2 invfval.n . . . 4 𝑁 = (Inv‘𝐶)
3 invfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invfval.x . . . 4 (𝜑𝑋𝐵)
5 invfval.y . . . 4 (𝜑𝑌𝐵)
61, 2, 3, 4, 5invsym2 17105 . . 3 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
76fveq1d 6665 . 2 (𝜑 → ((𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)))
8 isoval.n . . . 4 𝐼 = (Iso‘𝐶)
91, 2, 3, 4, 5, 8invf1o 17111 . . 3 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))
10 invinv.f . . 3 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
11 f1ocnvfv1 7031 . . 3 (((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)
129, 10, 11syl2anc 587 . 2 (𝜑 → ((𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)
137, 12eqtr3d 2795 1 (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ◡ccnv 5527  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  Catccat 17006  Invcinv 17087  Isociso 17088 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-cat 17010  df-cid 17011  df-sect 17089  df-inv 17090  df-iso 17091 This theorem is referenced by: (None)
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