Theorem sscpwex 17753

Description: An analogue of pwex 5330 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 7402	. 2 ⊢ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽) ∈ V
2		brssc 17752	. . . 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 3448	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
5	4, 4	xpex 7709	. . . . . . . . . 10 ⊢ (𝑡 × 𝑡) ∈ V
6		fnex 7173	. . . . . . . . . 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 7858	. . . . . . . . 9 ⊢ (𝐽 ∈ V → ran 𝐽 ∈ V)
9		uniexg 7696	. . . . . . . . 9 ⊢ (ran 𝐽 ∈ V → ∪ ran 𝐽 ∈ V)
10		pwexg 5328	. . . . . . . . 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 6603	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → dom 𝐽 ∈ V)
15		ss2ixp 8860	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ 𝒫 ∪ ran 𝐽 → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) ⊆ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
16		fvssunirn 6873	. . . . . . . . . . . . 13 ⊢ (𝐽‘𝑥) ⊆ ∪ ran 𝐽
17	16	sspwi 4571	. . . . . . . . . . . 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 3050	. . . . . . . . . 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 3941	. . . . . . . . 9 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 ∪ ran 𝐽)
22		vex 3448	. . . . . . . . . 10 ⊢ ℎ ∈ V
23	22	elixpconst 8855	. . . . . . . . 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 4566	. . . . . . . . . . 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 5646	. . . . . . . . . 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 3981	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑠 ∈ 𝒫 𝑡 ∧ ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))) → (𝑠 × 𝑠) ⊆ dom 𝐽)
30		elpm2r 8795	. . . . . . . 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 3135	. . . . . 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 1930	. . . 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 4029	. 2 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ⊆ (𝒫 ∪ ran 𝐽 ↑pm dom 𝐽)
37	1, 36	ssexi 5272	1 ⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   class class class wbr 5102   × cxp 5629  dom cdm 5631  ran crn 5632   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↑_pm cpm 8777  Xcixp 8847   ⊆_cat cssc 17745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-pm 8779  df-ixp 8848  df-ssc 17748

This theorem is referenced by:  issubc  17773