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Theorem sscpwex 17527
Description: An analogue of pwex 5303 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 7308 . 2 (𝒫 ran 𝐽pm dom 𝐽) ∈ V
2 brssc 17526 . . . 4 (cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
3 simpl 483 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4 vex 3436 . . . . . . . . . . 11 𝑡 ∈ V
54, 4xpex 7603 . . . . . . . . . 10 (𝑡 × 𝑡) ∈ V
6 fnex 7093 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
73, 5, 6sylancl 586 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝐽 ∈ V)
8 rnexg 7751 . . . . . . . . 9 (𝐽 ∈ V → ran 𝐽 ∈ V)
9 uniexg 7593 . . . . . . . . 9 (ran 𝐽 ∈ V → ran 𝐽 ∈ V)
10 pwexg 5301 . . . . . . . . 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 6536 . . . . . . . . . 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, 5eqeltrdi 2847 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → dom 𝐽 ∈ V)
15 ss2ixp 8698 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
16 fvssunirn 6803 . . . . . . . . . . . . 13 (𝐽𝑥) ⊆ ran 𝐽
1716sspwi 4547 . . . . . . . . . . . 12 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽
1817a1i 11 . . . . . . . . . . 11 (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
1915, 18mprg 3078 . . . . . . . . . 10 X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽
20 simprr 770 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
2119, 20sselid 3919 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
22 vex 3436 . . . . . . . . . 10 ∈ V
2322elixpconst 8693 . . . . . . . . 9 (X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽:(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
2421, 23sylib 217 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → :(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
25 elpwi 4542 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑡𝑠𝑡)
2625ad2antrl 725 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝑠𝑡)
27 xpss12 5604 . . . . . . . . . 10 ((𝑠𝑡𝑠𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
2826, 26, 27syl2anc 584 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
2928, 13sseqtrrd 3962 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
30 elpm2r 8633 . . . . . . . 8 (((𝒫 ran 𝐽 ∈ V ∧ dom 𝐽 ∈ V) ∧ (:(𝑠 × 𝑠)⟶𝒫 ran 𝐽 ∧ (𝑠 × 𝑠) ⊆ dom 𝐽)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3111, 14, 24, 29, 30syl22anc 836 . . . . . . 7 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3231rexlimdvaa 3214 . . . . . 6 (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → ∈ (𝒫 ran 𝐽pm dom 𝐽)))
3332imp 407 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3433exlimiv 1933 . . . 4 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
352, 34sylbi 216 . . 3 (cat 𝐽 ∈ (𝒫 ran 𝐽pm dom 𝐽))
3635abssi 4003 . 2 {cat 𝐽} ⊆ (𝒫 ran 𝐽pm dom 𝐽)
371, 36ssexi 5246 1 {cat 𝐽} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065  Vcvv 3432  wss 3887  𝒫 cpw 4533   cuni 4839   class class class wbr 5074   × cxp 5587  dom cdm 5589  ran crn 5590   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  pm cpm 8616  Xcixp 8685  cat cssc 17519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pm 8618  df-ixp 8686  df-ssc 17522
This theorem is referenced by:  issubc  17550
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