Theorem sscpwex 16298

Description: An analogue of pwex 4774 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 6577	. 2 ⊢ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽) ∈ V
2		brssc 16297	. . . 4 ⊢ (ℎ ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
3		simpl 472	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4		vex 3176	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
5	4, 4	xpex 6860	. . . . . . . . . 10 ⊢ (𝑡 × 𝑡) ∈ V
6		fnex 6386	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
7	3, 5, 6	sylancl 693	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 ∈ V)
8		rnexg 6990	. . . . . . . . 9 ⊢ (𝐽 ∈ V → ran 𝐽 ∈ V)
9		uniexg 6853	. . . . . . . . 9 ⊢ (ran 𝐽 ∈ V → ∪ ran 𝐽 ∈ V)
10		pwexg 4776	. . . . . . . . 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 5904	. . . . . . . . . 10 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 = (𝑡 × 𝑡))
14	13, 5	syl6eqel 2696	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 ∈ V)
15		ss2ixp 7807	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽 → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
16		fvssunirn 6127	. . . . . . . . . . . . 13 ⊢ (𝐽‘𝑥) ⊆ ∪ ran 𝐽
17		sspwb 4844	. . . . . . . . . . . . 13 ⊢ ((𝐽‘𝑥) ⊆ ∪ ran 𝐽 ↔ 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽)
18	16, 17	mpbi 219	. . . . . . . . . . . 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 2910	. . . . . . . . . 10 ⊢ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽
21		simprr 792	. . . . . . . . . 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 3566	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
23		vex 3176	. . . . . . . . . 10 ⊢ ℎ ∈ V
24	23	elixpconst 7802	. . . . . . . . 9 ⊢ (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽 ↔ ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
25	22, 24	sylib 207	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
26		elpwi 4117	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑡 → 𝑠 ⊆ 𝑡)
27	26	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝑠 ⊆ 𝑡)
28		xpss12 5148	. . . . . . . . . 10 ⊢ ((𝑠 ⊆ 𝑡 ∧ 𝑠 ⊆ 𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
29	27, 27, 28	syl2anc 691	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
30	29, 13	sseqtr4d 3605	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
31		elpm2r 7761	. . . . . . . 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 1319	. . . . . . 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 3014	. . . . . 6 ⊢ (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)))
34	33	imp 444	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
35	34	exlimiv 1845	. . . 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 206	. . 3 ⊢ (ℎ ⊆cat 𝐽 → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
37	36	abssi 3640	. 2 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ⊆ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)
38	1, 37	ssexi 4731	1 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_pm cpm 7745  Xcixp 7794   ⊆_cat cssc 16290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-ixp 7795  df-ssc 16293

This theorem is referenced by:  issubc  16318