Theorem elfm 21751

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 21747	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))))
2	1	eleq2d 2687	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)))))
3		eqid 2622	. . . . 5 ⊢ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))
4	3	fbasrn 21688	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐶) → ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ∈ (fBas‘𝑋))
5	4	3comr 1273	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ∈ (fBas‘𝑋))
6		elfg 21675	. . 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 477	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
9		eqid 2622	. . . . . 6 ⊢ (𝐹 “ 𝑥) = (𝐹 “ 𝑥)
10		imaeq2 5462	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → (𝐹 “ 𝑡) = (𝐹 “ 𝑥))
11	10	eqeq2d 2632	. . . . . . 7 ⊢ (𝑡 = 𝑥 → ((𝐹 “ 𝑥) = (𝐹 “ 𝑡) ↔ (𝐹 “ 𝑥) = (𝐹 “ 𝑥)))
12	11	rspcev 3309	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐹 “ 𝑥) = (𝐹 “ 𝑥)) → ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡))
13	8, 9, 12	sylancl 694	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡))
14		simpl1 1064	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐶)
15		imassrn 5477	. . . . . . . 8 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
16		frn 6053	. . . . . . . . . 10 ⊢ (𝐹:𝑌⟶𝑋 → ran 𝐹 ⊆ 𝑋)
17	16	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran 𝐹 ⊆ 𝑋)
18	17	adantr 481	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → ran 𝐹 ⊆ 𝑋)
19	15, 18	syl5ss 3614	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ⊆ 𝑋)
20	14, 19	ssexd 4805	. . . . . 6 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ V)
21		eqid 2622	. . . . . . 7 ⊢ (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))
22	21	elrnmpt 5372	. . . . . 6 ⊢ ((𝐹 “ 𝑥) ∈ V → ((𝐹 “ 𝑥) ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ↔ ∃𝑡 ∈ 𝐵 (𝐹 “ 𝑥) = (𝐹 “ 𝑡)))
23	20, 22	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)))
24	13, 23	mpbird 247	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)))
25	10	cbvmptv 4750	. . . . . . 7 ⊢ (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) = (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))
26	25	elrnmpt 5372	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) → (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥)))
27	26	ibi 256	. . . . 5 ⊢ (𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥))
28	27	adantl 482	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹 “ 𝑥))
29		simpr 477	. . . . 5 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 = (𝐹 “ 𝑥)) → 𝑦 = (𝐹 “ 𝑥))
30	29	sseq1d 3632	. . . 4 ⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 = (𝐹 “ 𝑥)) → (𝑦 ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
31	24, 28, 30	rexxfrd 4881	. . 3 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ ran (𝑡 ∈ 𝐵 ↦ (𝐹 “ 𝑡))𝑦 ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴))
32	31	anbi2d 740	. 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)))
33	2, 7, 32	3bitrd 294	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574   ↦ cmpt 4729  ran crn 5115   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  fBascfbas 19734  filGencfg 19735   FilMap cfm 21737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fm 21742

This theorem is referenced by:  elfm2  21752  fmfg  21753  rnelfm  21757  fmfnfmlem1  21758  fmfnfm  21762  fmco  21765  flfnei  21795  isflf  21797  isfcf  21838  filnetlem4  32376