Theorem sscpwex 17079

Description: An analogue of pwex 5273 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	⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Distinct variable group:   ℎ,𝐽

Proof of Theorem sscpwex

Dummy variables 𝑠 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7183	. 2 ⊢ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽) ∈ V
2		brssc 17078	. . . 4 ⊢ (ℎ ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
3		simpl 485	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4		vex 3497	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
5	4, 4	xpex 7470	. . . . . . . . . 10 ⊢ (𝑡 × 𝑡) ∈ V
6		fnex 6974	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
7	3, 5, 6	sylancl 588	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 ∈ V)
8		rnexg 7608	. . . . . . . . 9 ⊢ (𝐽 ∈ V → ran 𝐽 ∈ V)
9		uniexg 7460	. . . . . . . . 9 ⊢ (ran 𝐽 ∈ V → ∪ ran 𝐽 ∈ V)
10		pwexg 5271	. . . . . . . . 9 ⊢ (∪ ran 𝐽 ∈ V → 𝒫 ∪ ran 𝐽 ∈ V)
11	7, 8, 9, 10	4syl 19	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝒫 ∪ ran 𝐽 ∈ V)
12		fndm 6449	. . . . . . . . . 10 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
13	12	adantr 483	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 = (𝑡 × 𝑡))
14	13, 5	eqeltrdi 2921	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 ∈ V)
15		ss2ixp 8468	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽 → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
16		fvssunirn 6693	. . . . . . . . . . . . 13 ⊢ (𝐽‘𝑥) ⊆ ∪ ran 𝐽
17		sspwb 5333	. . . . . . . . . . . . 13 ⊢ ((𝐽‘𝑥) ⊆ ∪ ran 𝐽 ↔ 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽)
18	16, 17	mpbi 232	. . . . . . . . . . . 12 ⊢ 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽)
20	15, 19	mprg 3152	. . . . . . . . . 10 ⊢ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽
21		simprr 771	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))
22	20, 21	sseldi 3964	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
23		vex 3497	. . . . . . . . . 10 ⊢ ℎ ∈ V
24	23	elixpconst 8463	. . . . . . . . 9 ⊢ (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽 ↔ ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
25	22, 24	sylib 220	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
26		elpwi 4550	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑡 → 𝑠 ⊆ 𝑡)
27	26	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝑠 ⊆ 𝑡)
28		xpss12 5564	. . . . . . . . . 10 ⊢ ((𝑠 ⊆ 𝑡 ∧ 𝑠 ⊆ 𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
29	27, 27, 28	syl2anc 586	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
30	29, 13	sseqtrrd 4007	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
31		elpm2r 8418	. . . . . . . 8 ⊢ (((𝒫 ∪ ran 𝐽 ∈ V ∧ dom 𝐽 ∈ V) ∧ (ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽 ∧ (𝑠 × 𝑠) ⊆ dom 𝐽)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
32	11, 14, 25, 30, 31	syl22anc 836	. . . . . . 7 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
33	32	rexlimdvaa 3285	. . . . . 6 ⊢ (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)))
34	33	imp 409	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
35	34	exlimiv 1927	. . . 4 ⊢ (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
36	2, 35	sylbi 219	. . 3 ⊢ (ℎ ⊆cat 𝐽 → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
37	36	abssi 4045	. 2 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ⊆ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)
38	1, 37	ssexi 5218	1 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Syntax hints:   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  ∃wrex 3139  Vcvv 3494   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4831   class class class wbr 5058   × cxp 5547  dom cdm 5549  ran crn 5550   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ↑_pm cpm 8401  Xcixp 8455   ⊆_cat cssc 17071

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-ixp 8456  df-ssc 17074

This theorem is referenced by:  issubc  17099