Theorem elfm3 22560

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 6597	. . . 4 ⊢ (𝐹:𝑌–onto→𝑋 → (𝐹 “ 𝑌) = 𝑋)
2	1	adantl 484	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) = 𝑋)
3		fofun 6593	. . . 4 ⊢ (𝐹:𝑌–onto→𝑋 → Fun 𝐹)
4		elfvdm 6704	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5		funimaexg 6442	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom fBas) → (𝐹 “ 𝑌) ∈ V)
6	3, 4, 5	syl2anr 598	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) ∈ V)
7	2, 6	eqeltrrd 2916	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ∈ V)
8		fof 6592	. . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋)
9		elfm2.l	. . . . . 6 ⊢ 𝐿 = (𝑌filGen𝐵)
10	9	elfm2 22558	. . . . 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 1161	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
12		fgcl 22488	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
13	9, 12	eqeltrid 2919	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14	13	3ad2ant2 1130	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝐿 ∈ (Fil‘𝑌))
15	14	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16		simprl 769	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿)
17		cnvimass 5951	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
18		fofn 6594	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹 Fn 𝑌)
19		fndm 6457	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑌 → dom 𝐹 = 𝑌)
20	18, 19	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌–onto→𝑋 → dom 𝐹 = 𝑌)
21	17, 20	sseqtrid 4021	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → (◡𝐹 “ 𝐴) ⊆ 𝑌)
22	21	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
23	22	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ⊆ 𝑌)
24	3	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → Fun 𝐹)
25	24	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → Fun 𝐹)
26	9	eleq2i 2906	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐿 ↔ 𝑦 ∈ (𝑌filGen𝐵))
27		elfg 22481	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
28	27	3ad2ant2 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
29	28	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
30	26, 29	syl5bb 285	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ 𝐿 ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦)))
31	30	simprbda 501	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ 𝑌)
32		sseq2 3995	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹 ↔ 𝑦 ⊆ 𝑌))
33	32	biimpar 480	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝐹 = 𝑌 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
34	20, 33	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
35	34	3ad2antl3 1183	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
36	35	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹)
37	31, 36	syldan 593	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ dom 𝐹)
38		funimass3 6826	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ⊆ dom 𝐹) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
39	25, 37, 38	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴)))
40	39	biimpd 231	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 → 𝑦 ⊆ (◡𝐹 “ 𝐴)))
41	40	impr 457	. . . . . . . . 9 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (◡𝐹 “ 𝐴))
42		filss 22463	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦 ∈ 𝐿 ∧ (◡𝐹 “ 𝐴) ⊆ 𝑌 ∧ 𝑦 ⊆ (◡𝐹 “ 𝐴))) → (◡𝐹 “ 𝐴) ∈ 𝐿)
43	15, 16, 23, 41, 42	syl13anc 1368	. . . . . . . 8 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ∈ 𝐿)
44		foimacnv 6634	. . . . . . . . . . 11 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
45	44	eqcomd 2829	. . . . . . . . . 10 ⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
46	45	3ad2antl3 1183	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
47	46	adantr 483	. . . . . . . 8 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))
48		imaeq2 5927	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝐴)))
49	48	rspceeqv 3640	. . . . . . . 8 ⊢ (((◡𝐹 “ 𝐴) ∈ 𝐿 ∧ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))
50	43, 47, 49	syl2anc 586	. . . . . . 7 ⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))
51	50	rexlimdvaa 3287	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
52	51	expimpd 456	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
53		simprr 771	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 = (𝐹 “ 𝑥))
54		imassrn 5942	. . . . . . . . 9 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
55		forn 6595	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→𝑋 → ran 𝐹 = 𝑋)
56	55	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ran 𝐹 = 𝑋)
57	56	adantr 483	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ran 𝐹 = 𝑋)
58	54, 57	sseqtrid 4021	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐹 “ 𝑥) ⊆ 𝑋)
59	53, 58	eqsstrd 4007	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 ⊆ 𝑋)
60		eqimss2 4026	. . . . . . . . 9 ⊢ (𝐴 = (𝐹 “ 𝑥) → (𝐹 “ 𝑥) ⊆ 𝐴)
61		imaeq2 5927	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐹 “ 𝑦) = (𝐹 “ 𝑥))
62	61	sseq1d 4000	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
63	62	rspcev 3625	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐿 ∧ (𝐹 “ 𝑥) ⊆ 𝐴) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
64	60, 63	sylan2 594	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
65	64	adantl 484	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)
66	59, 65	jca 514	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))
67	66	rexlimdvaa 3287	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)))
68	52, 67	impbid 214	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
69	11, 68	bitrd 281	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
70	69	3coml 1123	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋 ∧ 𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))
71	7, 70	mpd3an3 1458	1 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  ^◡ccnv 5556  dom cdm 5557  ran crn 5558   “ cima 5560  Fun wfun 6351   Fn wfn 6352  ⟶wf 6353  –onto→wfo 6355  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  filGencfg 20536  Filcfil 22455   FilMap cfm 22543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-fbas 20544  df-fg 20545  df-fil 22456  df-fm 22548

This theorem is referenced by:  fmid  22570