Theorem elfm2 23007

Description: An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
elfm2.l	⊢ 𝐿 = (𝑌filGen𝐵)

Assertion

Ref	Expression
elfm2	⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝑋   𝑥,𝐴   𝑥,𝐿   𝑥,𝑌

Proof of Theorem elfm2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfm 23006	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
2		ssfg 22931	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
3		elfm2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑌filGen𝐵)
4	2, 3	sseqtrrdi 3968	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
5	4	sselda 3917	. . . . . . . 8 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐿)
6	5	adantrr 713	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
7	6	3ad2antl2 1184	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
8		simprr 769	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑦) ⊆ 𝐴)
9		imaeq2 5954	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10	9	sseq1d 3948	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ⊆ 𝐴 ↔ (𝐹 “ 𝑦) ⊆ 𝐴))
11	10	rspcev 3552	. . . . . 6 ⊢ ((𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
12	7, 8, 11	syl2anc 583	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
13	12	rexlimdvaa 3213	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
14	3	eleq2i 2830	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↔ 𝑥 ∈ (𝑌filGen𝐵))
15		elfg 22930	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ (𝑌filGen𝐵) ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
16	14, 15	syl5bb 282	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
17	16	3ad2ant2 1132	. . . . . 6 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
18		imass2 5999	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥))
19		sstr2 3924	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝐴 → (𝐹 “ 𝑦) ⊆ 𝐴))
20	19	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑥) ⊆ 𝐴 → ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → (𝐹 “ 𝑦) ⊆ 𝐴))
21	20	ad2antll 725	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → (𝐹 “ 𝑦) ⊆ 𝐴))
22	18, 21	syl5 34	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ 𝐴))
23	22	reximdv 3201	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))
24	23	expr 456	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ⊆ 𝑌) → ((𝐹 “ 𝑥) ⊆ 𝐴 → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
25	24	com23 86	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
26	25	expimpd 453	. . . . . 6 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
27	17, 26	sylbid 239	. . . . 5 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
28	27	rexlimdv 3211	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))
29	13, 28	impbid 211	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
30	29	anbi2d 628	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)))
31	1, 30	bitrd 278	1 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883   “ cima 5583  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  fBascfbas 20498  filGencfg 20499   FilMap cfm 22992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-fg 20508  df-fm 22997

This theorem is referenced by:  fmfg  23008  elfm3  23009  imaelfm  23010