Theorem sscpwex 17717

Description: An analogue of pwex 5313 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 7374	. 2 ⊢ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽) ∈ V
2		brssc 17716	. . . 4 ⊢ (ℎ ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
3		simpl 482	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 Fn (𝑡 × 𝑡))
4		vex 3440	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
5	4, 4	xpex 7681	. . . . . . . . . 10 ⊢ (𝑡 × 𝑡) ∈ V
6		fnex 7146	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
7	3, 5, 6	sylancl 586	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝐽 ∈ V)
8		rnexg 7827	. . . . . . . . 9 ⊢ (𝐽 ∈ V → ran 𝐽 ∈ V)
9		uniexg 7668	. . . . . . . . 9 ⊢ (ran 𝐽 ∈ V → ∪ ran 𝐽 ∈ V)
10		pwexg 5311	. . . . . . . . 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 6579	. . . . . . . . . 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	eqeltrdi 2839	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 ∈ V)
15		ss2ixp 8829	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽 → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
16		fvssunirn 6848	. . . . . . . . . . . . 13 ⊢ (𝐽‘𝑥) ⊆ ∪ ran 𝐽
17	16	sspwi 4557	. . . . . . . . . . . 12 ⊢ 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑠 × 𝑠) → 𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽)
19	15, 18	mprg 3053	. . . . . . . . . 10 ⊢ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽
20		simprr 772	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))
21	19, 20	sselid 3927	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
22		vex 3440	. . . . . . . . . 10 ⊢ ℎ ∈ V
23	22	elixpconst 8824	. . . . . . . . 9 ⊢ (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽 ↔ ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
24	21, 23	sylib 218	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ:(𝑠 × 𝑠)⟶𝒫 ∪ ran 𝐽)
25		elpwi 4552	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑡 → 𝑠 ⊆ 𝑡)
26	25	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → 𝑠 ⊆ 𝑡)
27		xpss12 5626	. . . . . . . . . 10 ⊢ ((𝑠 ⊆ 𝑡 ∧ 𝑠 ⊆ 𝑡) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
28	26, 26, 27	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ (𝑡 × 𝑡))
29	28, 13	sseqtrrd 3967	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
30		elpm2r 8764	. . . . . . . 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))
31	11, 14, 24, 29, 30	syl22anc 838	. . . . . . 7 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
32	31	rexlimdvaa 3134	. . . . . 6 ⊢ (𝐽 Fn (𝑡 × 𝑡) → (∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)))
33	32	imp 406	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
34	33	exlimiv 1931	. . . 4 ⊢ (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
35	2, 34	sylbi 217	. . 3 ⊢ (ℎ ⊆cat 𝐽 → ℎ ∈ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽))
36	35	abssi 4015	. 2 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ⊆ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)
37	1, 36	ssexi 5255	1 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854   class class class wbr 5086   × cxp 5609  dom cdm 5611  ran crn 5612   Fn wfn 6471  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ↑_pm cpm 8746  Xcixp 8816   ⊆_cat cssc 17709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-pm 8748  df-ixp 8817  df-ssc 17712

This theorem is referenced by:  issubc  17737