Theorem elfm3 23009

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 6677	. . . 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 6673	. . . 4 ⊢ (𝐹:𝑌–onto→𝑋 → Fun 𝐹)
4		elfvdm 6788	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5		funimaexg 6504	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom fBas) → (𝐹 “ 𝑌) ∈ V)
6	3, 4, 5	syl2anr 596	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) ∈ V)
7	2, 6	eqeltrrd 2840	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ∈ V)
8		fof 6672	. . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋)
9		elfm2.l	. . . . . 6 ⊢ 𝐿 = (𝑌filGen𝐵)
10	9	elfm2 23007	. . . . 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 1163	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
12		fgcl 22937	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
13	9, 12	eqeltrid 2843	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14	13	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝐿 ∈ (Fil‘𝑌))
15	14	ad2antrr 722	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16		simprl 767	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
17		cnvimass 5978	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
18		fofn 6674	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹 Fn 𝑌)
19	18	fndmd 6522	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌–onto→𝑋 → dom 𝐹 = 𝑌)
20	17, 19	sseqtrid 3969	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → (◡𝐹 “ 𝐴) ⊆ 𝑌)
21	20	3ad2ant3 1133	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
22	21	ad2antrr 722	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
23	3	3ad2ant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → Fun 𝐹)
24	23	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → Fun 𝐹)
25	9	eleq2i 2830	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐿 ↔ 𝑦 ∈ (𝑌filGen𝐵))
26		elfg 22930	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
27	26	3ad2ant2 1132	. . . . . . . . . . . . . . . 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	syl5bb 282	. . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
34	33	3ad2antl3 1185	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
35	34	adantlr 711	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
36	30, 35	syldan 590	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ dom 𝐹)
37		funimass3 6913	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ⊆ dom 𝐹) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
38	24, 36, 37	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
39	38	biimpd 228	. . . . . . . . . 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 22912	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦 ∈ 𝐿 ∧ (◡𝐹 “ 𝐴) ⊆ 𝑌 ∧ 𝑦 ⊆ (◡𝐹 “ 𝐴))) → (◡𝐹 “ 𝐴) ∈ 𝐿)
42	15, 16, 22, 40, 41	syl13anc 1370	. . . . . . . 8 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ∈ 𝐿)
43		foimacnv 6717	. . . . . . . . . . 11 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
44	43	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
45	44	3ad2antl3 1185	. . . . . . . . 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 5954	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝐴)))
48	47	rspceeqv 3567	. . . . . . . 8 ⊢ (((◡𝐹 “ 𝐴) ∈ 𝐿 ∧ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))
49	42, 46, 48	syl2anc 583	. . . . . . 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 3213	. . . . . 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 769	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 = (𝐹 “ 𝑥))
53		imassrn 5969	. . . . . . . . 9 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
54		forn 6675	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → ran 𝐹 = 𝑋)
55	54	3ad2ant3 1133	. . . . . . . . . 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 3969	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐹 “ 𝑥) ⊆ 𝑋)
58	52, 57	eqsstrd 3955	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 ⊆ 𝑋)
59		eqimss2 3974	. . . . . . . . 9 ⊢ (𝐴 = (𝐹 “ 𝑥) → (𝐹 “ 𝑥) ⊆ 𝐴)
60		imaeq2 5954	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐹 “ 𝑦) = (𝐹 “ 𝑥))
61	60	sseq1d 3948	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
62	61	rspcev 3552	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐿 ∧ (𝐹 “ 𝑥) ⊆ 𝐴) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
63	59, 62	sylan2 592	. . . . . . . 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 3213	. . . . 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 211	. . . 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 278	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
69	68	3coml 1125	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋 ∧ 𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
70	7, 69	mpd3an3 1460	1 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Fun wfun 6412  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  fBascfbas 20498  filGencfg 20499  Filcfil 22904   FilMap cfm 22992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-fg 20508  df-fil 22905  df-fm 22997

This theorem is referenced by:  fmid  23019