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Theorem sscpwex 17863
Description: An analogue of pwex 5386 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 7464 . 2 (𝒫 ran 𝐽pm dom 𝐽) ∈ V
2 brssc 17862 . . . 4 (cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
3 simpl 482 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4 vex 3482 . . . . . . . . . . 11 𝑡 ∈ V
54, 4xpex 7772 . . . . . . . . . 10 (𝑡 × 𝑡) ∈ V
6 fnex 7237 . . . . . . . . . 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 7925 . . . . . . . . 9 (𝐽 ∈ V → ran 𝐽 ∈ V)
9 uniexg 7759 . . . . . . . . 9 (ran 𝐽 ∈ V → ran 𝐽 ∈ V)
10 pwexg 5384 . . . . . . . . 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 6672 . . . . . . . . . 10 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
1312adantr 480 . . . . . . . . 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 8949 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
16 fvssunirn 6940 . . . . . . . . . . . . 13 (𝐽𝑥) ⊆ ran 𝐽
1716sspwi 4617 . . . . . . . . . . . 12 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽
1817a1i 11 . . . . . . . . . . 11 (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
1915, 18mprg 3065 . . . . . . . . . 10 X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽
20 simprr 773 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
2119, 20sselid 3993 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
22 vex 3482 . . . . . . . . . 10 ∈ V
2322elixpconst 8944 . . . . . . . . 9 (X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽:(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
2421, 23sylib 218 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → :(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
25 elpwi 4612 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑡𝑠𝑡)
2625ad2antrl 728 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝑠𝑡)
27 xpss12 5704 . . . . . . . . . 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 4037 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
30 elpm2r 8884 . . . . . . . 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 839 . . . . . . 7 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3231rexlimdvaa 3154 . . . . . 6 (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → ∈ (𝒫 ran 𝐽pm dom 𝐽)))
3332imp 406 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3433exlimiv 1928 . . . 4 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
352, 34sylbi 217 . . 3 (cat 𝐽 ∈ (𝒫 ran 𝐽pm dom 𝐽))
3635abssi 4080 . 2 {cat 𝐽} ⊆ (𝒫 ran 𝐽pm dom 𝐽)
371, 36ssexi 5328 1 {cat 𝐽} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   class class class wbr 5148   × cxp 5687  dom cdm 5689  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  pm cpm 8866  Xcixp 8936  cat cssc 17855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8868  df-ixp 8937  df-ssc 17858
This theorem is referenced by:  issubc  17886
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