Theorem elfm 23912

Description: An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

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

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

Proof of Theorem elfm

Dummy variables 𝑡 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmval 23908	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))))
2	1	eleq2d 2822	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)))))
3		eqid 2736	. . . . 5 ⊢ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))
4	3	fbasrn 23849	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐶) → ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ∈ (fBas‘𝑋))
5	4	3comr 1126	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ∈ (fBas‘𝑋))
6		elfg 23836	. . 3 ⊢ (ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))𝑦 ⊆ 𝐴)))
7	5, 6	syl 17	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))𝑦 ⊆ 𝐴)))
8		simpr 484	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
9		eqid 2736	. . . . . 6 ⊢ (𝐹 “ 𝑥) = (𝐹 “ 𝑥)
10		imaeq2 6021	. . . . . . 7 ⊢ (𝑡 = 𝑥 → (𝐹 “ 𝑡) = (𝐹 “ 𝑥))
11	10	rspceeqv 3587	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐹 “ 𝑥) = (𝐹 “ 𝑥)) → ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡))
12	8, 9, 11	sylancl 587	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡))
13		simpl1 1193	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐶)
14		imassrn 6036	. . . . . . . 8 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
15		frn 6675	. . . . . . . . . 10 ⊢ (𝐹:𝑌⟶𝑋 → ran 𝐹 ⊆ 𝑋)
16	15	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran 𝐹 ⊆ 𝑋)
17	16	adantr 480	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → ran 𝐹 ⊆ 𝑋)
18	14, 17	sstrid 3933	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ⊆ 𝑋)
19	13, 18	ssexd 5265	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ V)
20		eqid 2736	. . . . . . 7 ⊢ (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))
21	20	elrnmpt 5913	. . . . . 6 ⊢ ((𝐹 “ 𝑥) ∈ V → ((𝐹 “ 𝑥) ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ↔ ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡)))
22	19, 21	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → ((𝐹 “ 𝑥) ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ↔ ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡)))
23	12, 22	mpbird 257	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)))
24	10	cbvmptv 5189	. . . . . . 7 ⊢ (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))
25	24	elrnmpt 5913	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) → (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥)))
26	25	ibi 267	. . . . 5 ⊢ (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥))
27	26	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥))
28		simpr 484	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 = (𝐹 “ 𝑥)) → 𝑦 = (𝐹 “ 𝑥))
29	28	sseq1d 3953	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 = (𝐹 “ 𝑥)) → (𝑦 ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
30	23, 27, 29	rexxfrd 5351	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))𝑦 ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴))
31	30	anbi2d 631	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))𝑦 ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴)))
32	2, 7, 31	3bitrd 305	1 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889   ↦ cmpt 5166  ran crn 5632   “ 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:  elfm2  23913  fmfg  23914  rnelfm  23918  fmfnfmlem1  23919  fmfnfm  23923  fmco  23926  flfnei  23956  isflf  23958  isfcf  23999  filnetlem4  36563