MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sscpwex Structured version   Visualization version   GIF version

Theorem sscpwex 17073
Description: An analogue of pwex 5272 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 7178 . 2 (𝒫 ran 𝐽pm dom 𝐽) ∈ V
2 brssc 17072 . . . 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 3495 . . . . . . . . . . 11 𝑡 ∈ V
54, 4xpex 7465 . . . . . . . . . 10 (𝑡 × 𝑡) ∈ V
6 fnex 6971 . . . . . . . . . 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 7603 . . . . . . . . 9 (𝐽 ∈ V → ran 𝐽 ∈ V)
9 uniexg 7456 . . . . . . . . 9 (ran 𝐽 ∈ V → ran 𝐽 ∈ V)
10 pwexg 5270 . . . . . . . . 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 6448 . . . . . . . . . 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 2918 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → dom 𝐽 ∈ V)
15 ss2ixp 8462 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
16 fvssunirn 6692 . . . . . . . . . . . . 13 (𝐽𝑥) ⊆ ran 𝐽
17 sspwb 5332 . . . . . . . . . . . . 13 ((𝐽𝑥) ⊆ ran 𝐽 ↔ 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
1816, 17mpbi 231 . . . . . . . . . . . 12 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽
1918a1i 11 . . . . . . . . . . 11 (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽𝑥) ⊆ 𝒫 ran 𝐽)
2015, 19mprg 3149 . . . . . . . . . 10 X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽
21 simprr 769 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))
2220, 21sseldi 3962 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽)
23 vex 3495 . . . . . . . . . 10 ∈ V
2423elixpconst 8457 . . . . . . . . 9 (X𝑥 ∈ (𝑠 × 𝑠)𝒫 ran 𝐽:(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
2522, 24sylib 219 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → :(𝑠 × 𝑠)⟶𝒫 ran 𝐽)
26 elpwi 4547 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑡𝑠𝑡)
2726ad2antrl 724 . . . . . . . . . 10 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → 𝑠𝑡)
28 xpss12 5563 . . . . . . . . . 10 ((𝑠𝑡𝑠𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
2927, 27, 28syl2anc 584 . . . . . . . . 9 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
3029, 13sseqtrrd 4005 . . . . . . . 8 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
31 elpm2r 8413 . . . . . . . 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 834 . . . . . . 7 ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥))) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3332rexlimdvaa 3282 . . . . . 6 (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → ∈ (𝒫 ran 𝐽pm dom 𝐽)))
3433imp 407 . . . . 5 ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
3534exlimiv 1922 . . . 4 (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → ∈ (𝒫 ran 𝐽pm dom 𝐽))
362, 35sylbi 218 . . 3 (cat 𝐽 ∈ (𝒫 ran 𝐽pm dom 𝐽))
3736abssi 4043 . 2 {cat 𝐽} ⊆ (𝒫 ran 𝐽pm dom 𝐽)
381, 37ssexi 5217 1 {cat 𝐽} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057   × cxp 5546  dom cdm 5548  ran crn 5549   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  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-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8398  df-ixp 8450  df-ssc 17068
This theorem is referenced by:  issubc  17093
  Copyright terms: Public domain W3C validator