Theorem brssc 17858

Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
brssc	⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))

Distinct variable groups:   𝑡,𝑠,𝑥,𝐻   𝐽,𝑠,𝑡,𝑥

Proof of Theorem brssc

Dummy variables ℎ 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sscrel 17857	. . 3 ⊢ Rel ⊆cat
2	1	brrelex12i 5740	. 2 ⊢ (𝐻 ⊆cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3		vex 3484	. . . . . 6 ⊢ 𝑡 ∈ V
4	3, 3	xpex 7773	. . . . 5 ⊢ (𝑡 × 𝑡) ∈ V
5		fnex 7237	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
6	4, 5	mpan2 691	. . . 4 ⊢ (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7		elex 3501	. . . . 5 ⊢ (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
8	7	rexlimivw 3151	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
9	6, 8	anim12ci 614	. . 3 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
10	9	exlimiv 1930	. 2 ⊢ (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11		simpr 484	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
12	11	fneq1d 6661	. . . . 5 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13		simpl 482	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
14	11	fveq1d 6908	. . . . . . . . 9 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗‘𝑥) = (𝐽‘𝑥))
15	14	pweqd 4617	. . . . . . . 8 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝒫 (𝑗‘𝑥) = 𝒫 (𝐽‘𝑥))
16	15	ixpeq2dv 8953	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))
17	13, 16	eleq12d 2835	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) ↔ 𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
18	17	rexbidv 3179	. . . . 5 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) ↔ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
19	12, 18	anbi12d 632	. . . 4 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → ((𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥)) ↔ (𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))))
20	19	exbidv 1921	. . 3 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥)) ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))))
21		df-ssc 17854	. . 3 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
22	20, 21	brabga 5539	. 2 ⊢ ((𝐻 ∈ V ∧ 𝐽 ∈ V) → (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))))
23	2, 10, 22	pm5.21nii 378	1 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683   Fn wfn 6556  ‘cfv 6561  Xcixp 8937   ⊆_cat cssc 17851

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ixp 8938  df-ssc 17854

This theorem is referenced by:  sscpwex  17859  sscfn1  17861  sscfn2  17862  isssc  17864