Theorem elfm2 23913

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 23912	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
2		ssfg 23837	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
3		elfm2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑌filGen𝐵)
4	2, 3	sseqtrrdi 3963	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
5	4	sselda 3921	. . . . . . . 8 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐿)
6	5	adantrr 718	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
7	6	3ad2antl2 1188	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
8		simprr 773	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑦) ⊆ 𝐴)
9		imaeq2 6021	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10	9	sseq1d 3953	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ⊆ 𝐴 ↔ (𝐹 “ 𝑦) ⊆ 𝐴))
11	10	rspcev 3564	. . . . . 6 ⊢ ((𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
12	7, 8, 11	syl2anc 585	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
13	12	rexlimdvaa 3139	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
14	3	eleq2i 2828	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↔ 𝑥 ∈ (𝑌filGen𝐵))
15		elfg 23836	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ (𝑌filGen𝐵) ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
16	14, 15	bitrid 283	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
17	16	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
18		imass2 6067	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥))
19		sstr2 3928	. . . . . . . . . . . . 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 730	. . . . . . . . . . 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 3152	. . . . . . . . 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 240	. . . . 5 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
28	27	rexlimdv 3136	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))
29	13, 28	impbid 212	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
30	29	anbi2d 631	. 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 279	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  fBascfbas 21340  filGencfg 21341   FilMap cfm 23898

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-fbas 21349  df-fg 21350  df-fm 23903

This theorem is referenced by:  fmfg  23914  elfm3  23915  imaelfm  23916