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Theorem funciso 17829
Description: The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funciso.b 𝐡 = (Baseβ€˜π·)
funciso.s 𝐼 = (Isoβ€˜π·)
funciso.t 𝐽 = (Isoβ€˜πΈ)
funciso.f (πœ‘ β†’ 𝐹(𝐷 Func 𝐸)𝐺)
funciso.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
funciso.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
funciso.m (πœ‘ β†’ 𝑀 ∈ (π‘‹πΌπ‘Œ))
Assertion
Ref Expression
funciso (πœ‘ β†’ ((π‘‹πΊπ‘Œ)β€˜π‘€) ∈ ((πΉβ€˜π‘‹)𝐽(πΉβ€˜π‘Œ)))

Proof of Theorem funciso
StepHypRef Expression
1 eqid 2731 . 2 (Baseβ€˜πΈ) = (Baseβ€˜πΈ)
2 eqid 2731 . 2 (Invβ€˜πΈ) = (Invβ€˜πΈ)
3 funciso.f . . . . 5 (πœ‘ β†’ 𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 5149 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 217 . . . 4 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 17818 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) β†’ (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
75, 6syl 17 . . 3 (πœ‘ β†’ (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
87simprd 495 . 2 (πœ‘ β†’ 𝐸 ∈ Cat)
9 funciso.b . . . 4 𝐡 = (Baseβ€˜π·)
109, 1, 3funcf1 17821 . . 3 (πœ‘ β†’ 𝐹:𝐡⟢(Baseβ€˜πΈ))
11 funciso.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
1210, 11ffvelcdmd 7087 . 2 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (Baseβ€˜πΈ))
13 funciso.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
1410, 13ffvelcdmd 7087 . 2 (πœ‘ β†’ (πΉβ€˜π‘Œ) ∈ (Baseβ€˜πΈ))
15 funciso.t . 2 𝐽 = (Isoβ€˜πΈ)
16 eqid 2731 . . 3 (Invβ€˜π·) = (Invβ€˜π·)
17 funciso.s . . . 4 𝐼 = (Isoβ€˜π·)
187simpld 494 . . . 4 (πœ‘ β†’ 𝐷 ∈ Cat)
19 funciso.m . . . 4 (πœ‘ β†’ 𝑀 ∈ (π‘‹πΌπ‘Œ))
209, 17, 16, 18, 11, 13, 19invisoinvr 17743 . . 3 (πœ‘ β†’ 𝑀(𝑋(Invβ€˜π·)π‘Œ)((𝑋(Invβ€˜π·)π‘Œ)β€˜π‘€))
219, 16, 2, 3, 11, 13, 20funcinv 17828 . 2 (πœ‘ β†’ ((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)(Invβ€˜πΈ)(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜((𝑋(Invβ€˜π·)π‘Œ)β€˜π‘€)))
221, 2, 8, 12, 14, 15, 21inviso1 17718 1 (πœ‘ β†’ ((π‘‹πΊπ‘Œ)β€˜π‘€) ∈ ((πΉβ€˜π‘‹)𝐽(πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4634   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  Catccat 17613  Invcinv 17697  Isociso 17698   Func cfunc 17809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-map 8825  df-ixp 8895  df-cat 17617  df-cid 17618  df-sect 17699  df-inv 17700  df-iso 17701  df-func 17813
This theorem is referenced by:  ffthiso  17885
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