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Theorem brssc 16687
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 16686 . . 3 Rel ⊆cat
2 brrelex12 5294 . . 3 ((Rel ⊆cat𝐻cat 𝐽) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
31, 2mpan 705 . 2 (𝐻cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
4 vex 3351 . . . . . 6 𝑡 ∈ V
54, 4xpex 7107 . . . . 5 (𝑡 × 𝑡) ∈ V
6 fnex 6623 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
75, 6mpan2 706 . . . 4 (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
8 elex 3360 . . . . 5 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
98rexlimivw 3175 . . . 4 (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
107, 9anim12ci 593 . . 3 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
1110exlimiv 2008 . 2 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
12 simpr 480 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → 𝑗 = 𝐽)
1312fneq1d 6120 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
14 simpl 475 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → = 𝐻)
1512fveq1d 6333 . . . . . . . . 9 (( = 𝐻𝑗 = 𝐽) → (𝑗𝑥) = (𝐽𝑥))
1615pweqd 4299 . . . . . . . 8 (( = 𝐻𝑗 = 𝐽) → 𝒫 (𝑗𝑥) = 𝒫 (𝐽𝑥))
1716ixpeq2dv 8076 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
1814, 17eleq12d 2842 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → (X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ 𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1918rexbidv 3198 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
2013, 19anbi12d 746 . . . 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))))
2120exbidv 2000 . . 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))))
22 df-ssc 16683 . . 3 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
2321, 22brabga 5121 . 2 ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
243, 11, 23pm5.21nii 367 1 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1629  wex 1850  wcel 2143  wrex 3060  Vcvv 3348  𝒫 cpw 4294   class class class wbr 4783   × cxp 5246  Rel wrel 5253   Fn wfn 6025  cfv 6030  Xcixp 8060  cat cssc 16680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ixp 8061  df-ssc 16683
This theorem is referenced by:  sscpwex  16688  sscfn1  16690  sscfn2  16691  isssc  16693
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