Theorem brssc 17615

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 17614	. . 3 ⊢ Rel ⊆cat
2	1	brrelex12i 5667	. 2 ⊢ (𝐻 ⊆cat 𝐽 → (𝐻 ∈ V ∧ 𝐽 ∈ V))
3		vex 3445	. . . . . 6 ⊢ 𝑡 ∈ V
4	3, 3	xpex 7657	. . . . 5 ⊢ (𝑡 × 𝑡) ∈ V
5		fnex 7143	. . . . 5 ⊢ ((𝐽 Fn (𝑡 × 𝑡) ∧ (𝑡 × 𝑡) ∈ V) → 𝐽 ∈ V)
6	4, 5	mpan2 688	. . . 4 ⊢ (𝐽 Fn (𝑡 × 𝑡) → 𝐽 ∈ V)
7		elex 3459	. . . . 5 ⊢ (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 ∈ V)
8	7	rexlimivw 3144	. . . 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 1932	. 2 ⊢ (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → (𝐻 ∈ V ∧ 𝐽 ∈ V))
11		simpr 485	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
12	11	fneq1d 6572	. . . . 5 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑡 × 𝑡)))
13		simpl 483	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
14	11	fveq1d 6821	. . . . . . . . 9 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (𝑗‘𝑥) = (𝐽‘𝑥))
15	14	pweqd 4563	. . . . . . . 8 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → 𝒫 (𝑗‘𝑥) = 𝒫 (𝐽‘𝑥))
16	15	ixpeq2dv 8764	. . . . . . 7 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) = X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥))
17	13, 16	eleq12d 2831	. . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑗 = 𝐽) → (ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥) ↔ 𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
18	17	rexbidv 3171	. . . . 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 631	. . . 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 1923	. . 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 17611	. . 3 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
22	20, 21	brabga 5472	. 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 379	1 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃wrex 3070  Vcvv 3441  𝒫 cpw 4546   class class class wbr 5089   × cxp 5612   Fn wfn 6468  ‘cfv 6473  Xcixp 8748   ⊆_cat cssc 17608

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ixp 8749  df-ssc 17611

This theorem is referenced by:  sscpwex  17616  sscfn1  17618  sscfn2  17619  isssc  17621