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Theorem brssc 17858
Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
brssc (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Distinct variable groups:   𝑡,𝑠,𝑥,𝐻   𝐽,𝑠,𝑡,𝑥

Proof of Theorem brssc
Dummy variables 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscrel 17857 . . 3 Rel ⊆cat
21brrelex12i 5740 . 2 (𝐻cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3 vex 3484 . . . . . 6 𝑡 ∈ V
43, 3xpex 7773 . . . . 5 (𝑡 × 𝑡) ∈ V
5 fnex 7237 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
64, 5mpan2 691 . . . 4 (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7 elex 3501 . . . . 5 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
87rexlimivw 3151 . . . 4 (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
96, 8anim12ci 614 . . 3 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
109exlimiv 1930 . 2 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11 simpr 484 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → 𝑗 = 𝐽)
1211fneq1d 6661 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13 simpl 482 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → = 𝐻)
1411fveq1d 6908 . . . . . . . . 9 (( = 𝐻𝑗 = 𝐽) → (𝑗𝑥) = (𝐽𝑥))
1514pweqd 4617 . . . . . . . 8 (( = 𝐻𝑗 = 𝐽) → 𝒫 (𝑗𝑥) = 𝒫 (𝐽𝑥))
1615ixpeq2dv 8953 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
1713, 16eleq12d 2835 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → (X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ 𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1817rexbidv 3179 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1912, 18anbi12d 632 . . . 4 (( = 𝐻𝑗 = 𝐽) → ((𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥)) ↔ (𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
2019exbidv 1921 . . 3 (( = 𝐻𝑗 = 𝐽) → (∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥)) ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
21 df-ssc 17854 . . 3 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
2220, 21brabga 5539 . 2 ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
232, 10, 22pm5.21nii 378 1 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683   Fn wfn 6556  cfv 6561  Xcixp 8937  cat cssc 17851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ixp 8938  df-ssc 17854
This theorem is referenced by:  sscpwex  17859  sscfn1  17861  sscfn2  17862  isssc  17864
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