Theorem brssc 17776

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 17775	. . 3 ⊢ Rel ⊆cat
2	1	brrelex12i 5676	. 2 ⊢ (𝐻 ⊆cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3		vex 3437	. . . . . 6 ⊢ 𝑡 ∈ V
4	3, 3	xpex 7700	. . . . 5 ⊢ (𝑡 × 𝑡) ∈ V
5		fnex 7165	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
6	4, 5	mpan2 698	. . . 4 ⊢ (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7		elex 3454	. . . . 5 ⊢ (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
8	7	rexlimivw 3138	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
9	6, 8	anim12ci 621	. . 3 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
10	9	exlimiv 1938	. 2 ⊢ (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11		simpr 486	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
12	11	fneq1d 6582	. . . . 5 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13		simpl 484	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
14	11	fveq1d 6833	. . . . . . . . 9 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗‘𝑥) = (𝐽‘𝑥))
15	14	pweqd 4549	. . . . . . . 8 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝒫 (𝑗‘𝑥) = 𝒫 (𝐽‘𝑥))
16	15	ixpeq2dv 8855	. . . . . . 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 3165	. . . . 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 639	. . . 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 1929	. . 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 17772	. . 3 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
22	20, 21	brabga 5479	. 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 380	1 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃wrex 3065  Vcvv 3433  𝒫 cpw 4532   class class class wbr 5075   × cxp 5619   Fn wfn 6484  ‘cfv 6489  Xcixp 8839   ⊆_cat cssc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ixp 8840  df-ssc 17772

This theorem is referenced by:  sscpwex  17777  sscfn1  17779  sscfn2  17780  isssc  17782