Theorem fgss 24030

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 4007	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡))
2	1	anim2d 623	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
3	2	3ad2ant3 1153	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
4		elfg 24028	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
5	4	3ad2ant1 1151	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
6		elfg 24028	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
7	6	3ad2ant2 1152	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡)))
8	3, 5, 7	3imtr4d 297	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
9	8	ssrdv 3943	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  ∃wrex 3089   ⊆ wss 3905  ‘cfv 6536  (class class class)co 7410  fBascfbas 21510  filGencfg 21511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fg 21520

This theorem is referenced by:  fgabs  24036  fgtr  24047  fmss  24103