Theorem sscpwex 16827

Description: An analogue of pwex 5080 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 6937	. 2 ⊢ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽) ∈ V
2		brssc 16826	. . . 4 ⊢ (ℎ ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
3		simpl 476	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4		vex 3417	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
5	4, 4	xpex 7223	. . . . . . . . . 10 ⊢ (𝑡 × 𝑡) ∈ V
6		fnex 6737	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
7	3, 5, 6	sylancl 580	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 ∈ V)
8		rnexg 7359	. . . . . . . . 9 ⊢ (𝐽 ∈ V → ran 𝐽 ∈ V)
9		uniexg 7215	. . . . . . . . 9 ⊢ (ran 𝐽 ∈ V → ∪ ran 𝐽 ∈ V)
10		pwexg 5078	. . . . . . . . 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 6223	. . . . . . . . . 10 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
13	12	adantr 474	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 = (𝑡 × 𝑡))
14	13, 5	syl6eqel 2914	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 ∈ V)
15		ss2ixp 8188	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽 → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
16		fvssunirn 6462	. . . . . . . . . . . . 13 ⊢ (𝐽‘𝑥) ⊆ ∪ ran 𝐽
17		sspwb 5138	. . . . . . . . . . . . 13 ⊢ ((𝐽‘𝑥) ⊆ ∪ ran 𝐽 ↔ 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽)
18	16, 17	mpbi 222	. . . . . . . . . . . 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 3135	. . . . . . . . . 10 ⊢ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽
21		simprr 789	. . . . . . . . . 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 3825	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
23		vex 3417	. . . . . . . . . 10 ⊢ ℎ ∈ V
24	23	elixpconst 8183	. . . . . . . . 9 ⊢ (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽 ↔ ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
25	22, 24	sylib 210	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
26		elpwi 4388	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑡 → 𝑠 ⊆ 𝑡)
27	26	ad2antrl 719	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝑠 ⊆ 𝑡)
28		xpss12 5357	. . . . . . . . . 10 ⊢ ((𝑠 ⊆ 𝑡 ∧ 𝑠 ⊆ 𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
29	27, 27, 28	syl2anc 579	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
30	29, 13	sseqtr4d 3867	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
31		elpm2r 8140	. . . . . . . 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 872	. . . . . . 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 3241	. . . . . 6 ⊢ (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)))
34	33	imp 397	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
35	34	exlimiv 2029	. . . 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 209	. . 3 ⊢ (ℎ ⊆cat 𝐽 → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
37	36	abssi 3902	. 2 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ⊆ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)
38	1, 37	ssexi 5028	1 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Syntax hints:   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  {cab 2811  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873   × cxp 5340  dom cdm 5342  ran crn 5343   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ↑_pm cpm 8123  Xcixp 8175   ⊆_cat cssc 16819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-pm 8125  df-ixp 8176  df-ssc 16822

This theorem is referenced by:  issubc  16847