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Theorem brssc 17319
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 17318 . . 3 Rel ⊆cat
21brrelex12i 5604 . 2 (𝐻cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3 vex 3412 . . . . . 6 𝑡 ∈ V
43, 3xpex 7538 . . . . 5 (𝑡 × 𝑡) ∈ V
5 fnex 7033 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
64, 5mpan2 691 . . . 4 (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7 elex 3426 . . . . 5 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
87rexlimivw 3201 . . . 4 (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
96, 8anim12ci 617 . . 3 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
109exlimiv 1938 . 2 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11 simpr 488 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → 𝑗 = 𝐽)
1211fneq1d 6472 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13 simpl 486 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → = 𝐻)
1411fveq1d 6719 . . . . . . . . 9 (( = 𝐻𝑗 = 𝐽) → (𝑗𝑥) = (𝐽𝑥))
1514pweqd 4532 . . . . . . . 8 (( = 𝐻𝑗 = 𝐽) → 𝒫 (𝑗𝑥) = 𝒫 (𝐽𝑥))
1615ixpeq2dv 8594 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
1713, 16eleq12d 2832 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → (X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ 𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1817rexbidv 3216 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1912, 18anbi12d 634 . . . 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 1929 . . 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 17315 . . 3 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
2220, 21brabga 5415 . 2 ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
232, 10, 22pm5.21nii 383 1 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wrex 3062  Vcvv 3408  𝒫 cpw 4513   class class class wbr 5053   × cxp 5549   Fn wfn 6375  cfv 6380  Xcixp 8578  cat cssc 17312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ixp 8579  df-ssc 17315
This theorem is referenced by:  sscpwex  17320  sscfn1  17322  sscfn2  17323  isssc  17325
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