Theorem sess1 4339

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 4048	. . . . 5 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → 𝑦𝑆𝑥))
3	2	ss2rabdv 3238	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥})
4		ssexg 4144	. . . . 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 2545	. 2 ⊢ (𝑅 ⊆ 𝑆 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
8		df-se 4335	. 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V)
9		df-se 4335	. 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 2148  ∀wral 2455  {crab 2459  Vcvv 2739   ⊆ wss 3131   class class class wbr 4005   Se wse 4331

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-in 3137  df-ss 3144  df-br 4006  df-se 4335

This theorem is referenced by:  seeq1  4341