Theorem sess1 4383

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

Assertion

Ref	Expression
sess1	⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))

Proof of Theorem sess1

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

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → 𝑅 ⊆ 𝑆)
2	1	ssbrd 4086	. . . . 5 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → 𝑦𝑆𝑥))
3	2	ss2rabdv 3273	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥})
4		ssexg 4182	. . . . 5 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
5	4	ex 115	. . . 4 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
6	3, 5	syl 14	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
7	6	ralimdv 2573	. 2 ⊢ (𝑅 ⊆ 𝑆 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
8		df-se 4379	. 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V)
9		df-se 4379	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	3imtr4g 205	1 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2175  ∀wral 2483  {crab 2487  Vcvv 2771   ⊆ wss 3165   class class class wbr 4043   Se wse 4375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-sep 4161

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rab 2492  df-v 2773  df-in 3171  df-ss 3178  df-br 4044  df-se 4379

This theorem is referenced by:  seeq1  4385