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Theorem brssc 17697
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 17696 . . 3 Rel ⊆cat
21brrelex12i 5687 . 2 (𝐻cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3 vex 3449 . . . . . 6 𝑡 ∈ V
43, 3xpex 7687 . . . . 5 (𝑡 × 𝑡) ∈ V
5 fnex 7167 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
64, 5mpan2 689 . . . 4 (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7 elex 3463 . . . . 5 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 ∈ V)
87rexlimivw 3148 . . . 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 1933 . 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 6595 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13 simpl 483 . . . . . . 7 (( = 𝐻𝑗 = 𝐽) → = 𝐻)
1411fveq1d 6844 . . . . . . . . 9 (( = 𝐻𝑗 = 𝐽) → (𝑗𝑥) = (𝐽𝑥))
1514pweqd 4577 . . . . . . . 8 (( = 𝐻𝑗 = 𝐽) → 𝒫 (𝑗𝑥) = 𝒫 (𝐽𝑥))
1615ixpeq2dv 8851 . . . . . . 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 3175 . . . . 5 (( = 𝐻𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
1912, 18anbi12d 631 . . . 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 1924 . . 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 17693 . . 3 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
2220, 21brabga 5491 . 2 ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))))
232, 10, 22pm5.21nii 379 1 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3073  Vcvv 3445  𝒫 cpw 4560   class class class wbr 5105   × cxp 5631   Fn wfn 6491  cfv 6496  Xcixp 8835  cat cssc 17690
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ixp 8836  df-ssc 17693
This theorem is referenced by:  sscpwex  17698  sscfn1  17700  sscfn2  17701  isssc  17703
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