Theorem elfm2 24279

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 24278	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
2		ssfg 24203	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
3		elfm2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑌filGen𝐵)
4	2, 3	sseqtrrdi 3953	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
5	4	sselda 3912	. . . . . . . 8 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐿)
6	5	adantrr 730	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
7	6	3ad2antl2 1205	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
8		simprr 785	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑦) ⊆ 𝐴)
9		imaeq2 6053	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10	9	sseq1d 3943	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ⊆ 𝐴 ↔ (𝐹 “ 𝑦) ⊆ 𝐴))
11	10	rspcev 3579	. . . . . 6 ⊢ ((𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
12	7, 8, 11	syl2anc 596	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
13	12	rexlimdvaa 3167	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
14	3	eleq2i 2855	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↔ 𝑥 ∈ (𝑌filGen𝐵))
15		elfg 24202	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ (𝑌filGen𝐵) ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
16	14, 15	bitrid 286	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
17	16	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
18		imass2 6060	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥))
19		sstr2 3919	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝐴 → (𝐹 “ 𝑦) ⊆ 𝐴))
20	19	com12 33	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑥) ⊆ 𝐴 → ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → (𝐹 “ 𝑦) ⊆ 𝐴))
21	20	ad2antll 742	. . . . . . . . . . 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 35	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ 𝐴))
23	22	reximdv 3180	. . . . . . . . 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 462	. . . . . . . 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 87	. . . . . . 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 459	. . . . . 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 243	. . . . 5 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
28	27	rexlimdv 3164	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))
29	13, 28	impbid 215	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
30	29	anbi2d 642	. 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 282	1 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∃wrex 3089   ⊆ wss 3899   “ cima 5657  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427  fBascfbas 21678  filGencfg 21679   FilMap cfm 24264

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-fbas 21687  df-fg 21688  df-fm 24269

This theorem is used by:  fmfg  24280  elfm3  24281  imaelfm  24282