Theorem fgss 23767

Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgss	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Proof of Theorem fgss

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

Step	Hyp	Ref	Expression
1		ssrexv 4019	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡))
2	1	anim2d 612	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
3	2	3ad2ant3 1135	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
4		elfg 23765	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
6		elfg 23765	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
7	6	3ad2ant2 1134	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
8	3, 5, 7	3imtr4d 294	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
9	8	ssrdv 3955	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  fBascfbas 21259  filGencfg 21260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-fg 21269

This theorem is referenced by:  fgabs  23773  fgtr  23784  fmss  23840