Theorem ssc1 17083

Description: Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
isssc.1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
isssc.2	⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
ssc1.3	⊢ (𝜑 → 𝐻 ⊆cat 𝐽)

Assertion

Ref	Expression
ssc1	⊢ (𝜑 → 𝑆 ⊆ 𝑇)

Proof of Theorem ssc1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssc1.3	. . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
2		isssc.1	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
3		isssc.2	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
4		sscrel 17075	. . . . . . 7 ⊢ Rel ⊆cat
5	4	brrelex2i 5573	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
6	1, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
7	3	ssclem 17081	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V))
8	6, 7	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
9	2, 3, 8	isssc 17082	. . 3 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
10	1, 9	mpbid 235	. 2 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
11	10	simpld 498	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   × cxp 5517   Fn wfn 6319  (class class class)co 7135   ⊆_cat cssc 17069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-ixp 8445  df-ssc 17072

This theorem is referenced by:  ssctr  17087  ssceq  17088  subcss1  17104  issubc3  17111  subsubc  17115