Theorem brssc 17740

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 17739	. . 3 ⊢ Rel ⊆cat
2	1	brrelex12i 5679	. 2 ⊢ (𝐻 ⊆cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3		vex 3444	. . . . . 6 ⊢ 𝑡 ∈ V
4	3, 3	xpex 7698	. . . . 5 ⊢ (𝑡 × 𝑡) ∈ V
5		fnex 7163	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
6	4, 5	mpan2 691	. . . 4 ⊢ (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7		elex 3461	. . . . 5 ⊢ (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
8	7	rexlimivw 3133	. . . 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 1931	. 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 6585	. . . . 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 6836	. . . . . . . . 9 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗‘𝑥) = (𝐽‘𝑥))
15	14	pweqd 4571	. . . . . . . 8 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝒫 (𝑗‘𝑥) = 𝒫 (𝐽‘𝑥))
16	15	ixpeq2dv 8853	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))
17	13, 16	eleq12d 2830	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) ↔ 𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
18	17	rexbidv 3160	. . . . 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 1922	. . 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 17736	. . 3 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
22	20, 21	brabga 5482	. 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 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3060  Vcvv 3440  𝒫 cpw 4554   class class class wbr 5098   × cxp 5622   Fn wfn 6487  ‘cfv 6492  Xcixp 8837   ⊆_cat cssc 17733

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ixp 8838  df-ssc 17736

This theorem is referenced by:  sscpwex  17741  sscfn1  17743  sscfn2  17744  isssc  17746