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Theorem brcici 16774
Description: Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020.)
Hypotheses
Ref Expression
cic.i 𝐼 = (Iso‘𝐶)
cic.b 𝐵 = (Base‘𝐶)
cic.c (𝜑𝐶 ∈ Cat)
cic.x (𝜑𝑋𝐵)
cic.y (𝜑𝑌𝐵)
cic.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
brcici (𝜑𝑋( ≃𝑐𝐶)𝑌)

Proof of Theorem brcici
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cic.f . . 3 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
2 eleq1 2866 . . . 4 (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌)))
32spcegv 3482 . . 3 (𝐹 ∈ (𝑋𝐼𝑌) → (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)))
41, 1, 3sylc 65 . 2 (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))
5 cic.i . . 3 𝐼 = (Iso‘𝐶)
6 cic.b . . 3 𝐵 = (Base‘𝐶)
7 cic.c . . 3 (𝜑𝐶 ∈ Cat)
8 cic.x . . 3 (𝜑𝑋𝐵)
9 cic.y . . 3 (𝜑𝑌𝐵)
105, 6, 7, 8, 9cic 16773 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)))
114, 10mpbird 249 1 (𝜑𝑋( ≃𝑐𝐶)𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wex 1875  wcel 2157   class class class wbr 4843  cfv 6101  (class class class)co 6878  Basecbs 16184  Catccat 16639  Isociso 16720  𝑐 ccic 16769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-supp 7533  df-inv 16722  df-iso 16723  df-cic 16770
This theorem is referenced by:  cicref  16775  cicsym  16778  cictr  16779
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