Theorem elfm3 23813

Description: An alternate formulation of elementhood in a mapping filter that requires 𝐹 to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
elfm2.l	⊢ 𝐿 = (𝑌filGen𝐵)

Assertion

Ref	Expression
elfm3	⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))

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

Proof of Theorem elfm3

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

Step	Hyp	Ref	Expression
1		foima 6759	. . . 4 ⊢ (𝐹:𝑌–onto→𝑋 → (𝐹 “ 𝑌) = 𝑋)
2	1	adantl 481	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) = 𝑋)
3		fofun 6755	. . . 4 ⊢ (𝐹:𝑌–onto→𝑋 → Fun 𝐹)
4		elfvdm 6877	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5		funimaexg 6587	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom fBas) → (𝐹 “ 𝑌) ∈ V)
6	3, 4, 5	syl2anr 597	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) ∈ V)
7	2, 6	eqeltrrd 2829	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ∈ V)
8		fof 6754	. . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋)
9		elfm2.l	. . . . . 6 ⊢ 𝐿 = (𝑌filGen𝐵)
10	9	elfm2 23811	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
11	8, 10	syl3an3 1165	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
12		fgcl 23741	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
13	9, 12	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14	13	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝐿 ∈ (Fil‘𝑌))
15	14	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16		simprl 770	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
17		cnvimass 6042	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
18		fofn 6756	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹 Fn 𝑌)
19	18	fndmd 6605	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌–onto→𝑋 → dom 𝐹 = 𝑌)
20	17, 19	sseqtrid 3986	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → (◡𝐹 “ 𝐴) ⊆ 𝑌)
21	20	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
22	21	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
23	3	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → Fun 𝐹)
24	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → Fun 𝐹)
25	9	eleq2i 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐿 ↔ 𝑦 ∈ (𝑌filGen𝐵))
26		elfg 23734	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
27	26	3ad2ant2 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
28	27	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
29	25, 28	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ 𝐿 ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
30	29	simprbda 498	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ 𝑌)
31		sseq2 3970	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹 ↔ 𝑦 ⊆ 𝑌))
32	31	biimpar 477	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝐹 = 𝑌 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
33	19, 32	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
34	33	3ad2antl3 1188	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
35	34	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
36	30, 35	syldan 591	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ dom 𝐹)
37		funimass3 7008	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ⊆ dom 𝐹) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
38	24, 36, 37	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
39	38	biimpd 229	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 → 𝑦 ⊆ (◡𝐹 “ 𝐴)))
40	39	impr 454	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (◡𝐹 “ 𝐴))
41		filss 23716	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦 ∈ 𝐿 ∧ (◡𝐹 “ 𝐴) ⊆ 𝑌 ∧ 𝑦 ⊆ (◡𝐹 “ 𝐴))) → (◡𝐹 “ 𝐴) ∈ 𝐿)
42	15, 16, 22, 40, 41	syl13anc 1374	. . . . . . . 8 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ∈ 𝐿)
43		foimacnv 6799	. . . . . . . . . . 11 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
44	43	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
45	44	3ad2antl3 1188	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
46	45	adantr 480	. . . . . . . 8 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
47		imaeq2 6016	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝐴)))
48	47	rspceeqv 3608	. . . . . . . 8 ⊢ (((◡𝐹 “ 𝐴) ∈ 𝐿 ∧ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))
49	42, 46, 48	syl2anc 584	. . . . . . 7 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))
50	49	rexlimdvaa 3135	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
51	50	expimpd 453	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
52		simprr 772	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 = (𝐹 “ 𝑥))
53		imassrn 6031	. . . . . . . . 9 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
54		forn 6757	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → ran 𝐹 = 𝑋)
55	54	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ran 𝐹 = 𝑋)
56	55	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ran 𝐹 = 𝑋)
57	53, 56	sseqtrid 3986	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐹 “ 𝑥) ⊆ 𝑋)
58	52, 57	eqsstrd 3978	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 ⊆ 𝑋)
59		eqimss2 4003	. . . . . . . . 9 ⊢ (𝐴 = (𝐹 “ 𝑥) → (𝐹 “ 𝑥) ⊆ 𝐴)
60		imaeq2 6016	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐹 “ 𝑦) = (𝐹 “ 𝑥))
61	60	sseq1d 3975	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
62	61	rspcev 3585	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐿 ∧ (𝐹 “ 𝑥) ⊆ 𝐴) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
63	59, 62	sylan2 593	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
64	63	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
65	58, 64	jca 511	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))
66	65	rexlimdvaa 3135	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
67	51, 66	impbid 212	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
68	11, 67	bitrd 279	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
69	68	3coml 1127	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋 ∧ 𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
70	7, 69	mpd3an3 1464	1 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6493  ⟶wf 6495  –onto→wfo 6497  ‘cfv 6499  (class class class)co 7369  fBascfbas 21228  filGencfg 21229  Filcfil 23708   FilMap cfm 23796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-fbas 21237  df-fg 21238  df-fil 23709  df-fm 23801

This theorem is referenced by:  fmid  23823