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Theorem brcici 17068
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 2903 . . . 4 (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌)))
32spcegv 3583 . . 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 17067 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)))
114, 10mpbird 260 1 (𝜑𝑋( ≃𝑐𝐶)𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wex 1781  wcel 2115   class class class wbr 5053  cfv 6344  (class class class)co 7146  Basecbs 16481  Catccat 16933  Isociso 17014  𝑐 ccic 17063
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-supp 7823  df-inv 17016  df-iso 17017  df-cic 17064
This theorem is referenced by:  cicref  17069  cicsym  17072  cictr  17073
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