Theorem elfm2 23932

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 23931	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
2		ssfg 23856	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
3		elfm2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑌filGen𝐵)
4	2, 3	sseqtrrdi 3956	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
5	4	sselda 3915	. . . . . . . 8 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐿)
6	5	adantrr 723	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
7	6	3ad2antl2 1193	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
8		simprr 778	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑦) ⊆ 𝐴)
9		imaeq2 6009	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10	9	sseq1d 3946	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ⊆ 𝐴 ↔ (𝐹 “ 𝑦) ⊆ 𝐴))
11	10	rspcev 3560	. . . . . 6 ⊢ ((𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
12	7, 8, 11	syl2anc 590	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)
13	12	rexlimdvaa 3141	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
14	3	eleq2i 2831	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↔ 𝑥 ∈ (𝑌filGen𝐵))
15		elfg 23855	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ (𝑌filGen𝐵) ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
16	14, 15	bitrid 284	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
17	16	3ad2ant2 1140	. . . . . 6 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
18		imass2 6055	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥))
19		sstr2 3922	. . . . . . . . . . . . 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 735	. . . . . . . . . . 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 3154	. . . . . . . . 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 457	. . . . . . . 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 454	. . . . . 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 241	. . . . 5 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)))
28	27	rexlimdv 3138	. . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))
29	13, 28	impbid 213	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))
30	29	anbi2d 636	. 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 280	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883   “ cima 5622  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  fBascfbas 21336  filGencfg 21337   FilMap cfm 23917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 21345  df-fg 21346  df-fm 23922

This theorem is referenced by:  fmfg  23933  elfm3  23934  imaelfm  23935