Theorem fmfg 23930

Description: The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
elfm2.l	⊢ 𝐿 = (𝑌filGen𝐵)

Assertion

Ref	Expression
fmfg	⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))

Proof of Theorem fmfg

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

Step	Hyp	Ref	Expression
1		elfm2.l	. . . 4 ⊢ 𝐿 = (𝑌filGen𝐵)
2	1	elfm2 23929	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
3		fgcl 23859	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
4	1, 3	eqeltrid 2841	. . . . 5 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
5		filfbas 23829	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6	4, 5	syl 17	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
7		elfm 23928	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
8	6, 7	syl3an2 1165	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
9	2, 8	bitr4d 282	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))
10	9	eqrdv 2735	1 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890   “ cima 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7364  fBascfbas 21338  filGencfg 21339  Filcfil 23826   FilMap cfm 23914

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7367  df-oprab 7368  df-mpo 7369  df-fbas 21347  df-fg 21348  df-fil 23827  df-fm 23919

This theorem is referenced by:  fmfnfm  23939  cmetcaulem  25271