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Theorem sscpwex 16396
Description: An analogue of pwex 4808 for the subcategory subset relation: The collection of subcategory subsets of a given set 𝐽 is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
sscpwex {cat 𝐽} ∈ V
Distinct variable group:   ,𝐽

Proof of Theorem sscpwex
Dummy variables 𝑠 𝑡 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6632 . 2 (𝒫 ran 𝐽pm dom 𝐽) ∈ V
2 brssc 16395 . . . 4 (cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
3 simpl 473 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4 vex 3189 . . . . . . . . . . 11 𝑡 ∈ V
54, 4xpex 6915 . . . . . . . . . 10 (𝑡 × 𝑡) ∈ V
6 fnex 6435 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
73, 5, 6sylancl 693 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝐽 ∈ V)
8 rnexg 7045 . . . . . . . . 9 (𝐽 ∈ V → ran 𝐽 ∈ V)
9 uniexg 6908 . . . . . . . . 9 (ran 𝐽 ∈ V → ran 𝐽 ∈ V)
10 pwexg 4810 . . . . . . . . 9 ( ran 𝐽 ∈ V → 𝒫 ran 𝐽 ∈ V)
117, 8, 9, 104syl 19 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝒫 ran 𝐽 ∈ V)
12 fndm 5948 . . . . . . . . . 10 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
1312adantr 481 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → dom 𝐽 = (𝑡 × 𝑡))
1413, 5syl6eqel 2706 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → dom 𝐽 ∈ V)
15 ss2ixp 7865 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
16 fvssunirn 6174 . . . . . . . . . . . . 13 (𝐽𝑥) ⊆ ran 𝐽
17 sspwb 4878 . . . . . . . . . . . . 13 ((𝐽𝑥) ⊆ ran 𝐽 ↔ 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
1816, 17mpbi 220 . . . . . . . . . . . 12 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽
1918a1i 11 . . . . . . . . . . 11 (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
2015, 19mprg 2921 . . . . . . . . . 10 X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽
21 simprr 795 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
2220, 21sseldi 3581 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
23 vex 3189 . . . . . . . . . 10 ∈ V
2423elixpconst 7860 . . . . . . . . 9 (X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽:(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
2522, 24sylib 208 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → :(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
26 elpwi 4140 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑡𝑠𝑡)
2726ad2antrl 763 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝑠𝑡)
28 xpss12 5186 . . . . . . . . . 10 ((𝑠𝑡𝑠𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
2927, 27, 28syl2anc 692 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
3029, 13sseqtr4d 3621 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
31 elpm2r 7819 . . . . . . . 8 (((𝒫 ran 𝐽 ∈ V ∧ dom 𝐽 ∈ V) ∧ (:(𝑠 × 𝑠)⟶𝒫 ran 𝐽 ∧ (𝑠 × 𝑠) ⊆ dom 𝐽)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3211, 14, 25, 30, 31syl22anc 1324 . . . . . . 7 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3332rexlimdvaa 3025 . . . . . 6 (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → ∈ (𝒫 ran 𝐽pm dom 𝐽)))
3433imp 445 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3534exlimiv 1855 . . . 4 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
362, 35sylbi 207 . . 3 (cat 𝐽 ∈ (𝒫 ran 𝐽pm dom 𝐽))
3736abssi 3656 . 2 {cat 𝐽} ⊆ (𝒫 ran 𝐽pm dom 𝐽)
381, 37ssexi 4763 1 {cat 𝐽} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  Xcixp 7852  cat cssc 16388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pm 7805  df-ixp 7853  df-ssc 16391
This theorem is referenced by:  issubc  16416
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