Theorem sess2 5488

Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
sess2	⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴))

Proof of Theorem sess2

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

Step	Hyp	Ref	Expression
1		ssralv 3981	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V))
2		rabss2 4005	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥})
3		ssexg 5191	. . . . . 6 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∧ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	3	ex 416	. . . . 5 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
5	2, 4	syl 17	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
6	5	ralimdv 3145	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
7	1, 6	syld 47	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
8		df-se 5479	. 2 ⊢ (𝑅 Se 𝐵 ↔ ∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V)
9		df-se 5479	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	3imtr4g 299	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   Se wse 5476

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-sep 5167

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-se 5479

This theorem is referenced by:  seeq2  5492  wereu2  5516  wfrlem5  7942  frpomin  33191  frmin  33197  fprlem1  33250  frrlem15  33255