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Theorem brssc 17072
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 17071 . . 3 Rel ⊆cat
21brrelex12i 5600 . 2 (𝐻cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3 vex 3495 . . . . . 6 𝑡 ∈ V
43, 3xpex 7465 . . . . 5 (𝑡 × 𝑡) ∈ V
5 fnex 6971 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
64, 5mpan2 687 . . . 4 (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7 elex 3510 . . . . 5 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
87rexlimivw 3279 . . . 4 (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
96, 8anim12ci 613 . . 3 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
109exlimiv 1922 . 2 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11 simpr 485 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → 𝑗 = 𝐽)
1211fneq1d 6439 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13 simpl 483 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → = 𝐻)
1411fveq1d 6665 . . . . . . . . 9 (( = 𝐻𝑗 = 𝐽) → (𝑗𝑥) = (𝐽𝑥))
1514pweqd 4540 . . . . . . . 8 (( = 𝐻𝑗 = 𝐽) → 𝒫 (𝑗𝑥) = 𝒫 (𝐽𝑥))
1615ixpeq2dv 8465 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
1713, 16eleq12d 2904 . . . . . 6 (( = 𝐻𝑗 = 𝐽) → (X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ 𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1817rexbidv 3294 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1912, 18anbi12d 630 . . . 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 1913 . . 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 17068 . . 3 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
2220, 21brabga 5412 . 2 ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
232, 10, 22pm5.21nii 380 1 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492  𝒫 cpw 4535   class class class wbr 5057   × cxp 5546   Fn wfn 6343  cfv 6348  Xcixp 8449  cat cssc 17065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ixp 8450  df-ssc 17068
This theorem is referenced by:  sscpwex  17073  sscfn1  17075  sscfn2  17076  isssc  17078
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