Theorem brtrclfv2 42154

Description: Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)

Assertion

Ref	Expression
brtrclfv2	⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Distinct variable groups:   𝑅,𝑓   𝑈,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑌

Proof of Theorem brtrclfv2

Dummy variables 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5126	. . . 4 ⊢ (𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
2	1	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
3		trclfv 14912	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
4	3	breqd 5136	. . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
5	4	3ad2ant3 1135	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
6		elimasng 6060	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
7	6	3adant3 1132	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
8	2, 5, 7	3bitr4d 310	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋})))
9		intimasn 42084	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
10	9	3ad2ant1 1133	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
11		simpl3 1193	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ∈ 𝑊)
12		snex 5408	. . . . . . . . . . . . . . 15 ⊢ {𝑋} ∈ V
13		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
14	12, 13	xpex 7707	. . . . . . . . . . . . . 14 ⊢ ({𝑋} × 𝑓) ∈ V
15		unexg 7703	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
16	11, 14, 15	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17		trclfvlb 14920	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
18	17	unssad 4167	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
19	16, 18	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
20		trclfvcotrg 14928	. . . . . . . . . . . . 13 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
21	20	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
22		simpl1 1191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ 𝑈)
23		snidg 4640	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ {𝑋})
24	22, 23	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25		inelcm 4444	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
26	24, 24, 25	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27		xpima2 6156	. . . . . . . . . . . . . . 15 ⊢ (({𝑋} ∩ {𝑋}) ≠ ∅ → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
28	26, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
29	16, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
30	29	unssbd 4168	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31		imass1 6073	. . . . . . . . . . . . . . 15 ⊢ (({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
32	30, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
33	28, 32	eqsstrrd 4001	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34		imaundir 6123	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36		imassrn 6044	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37		rnxpss 6144	. . . . . . . . . . . . . . . . . 18 ⊢ ran ({𝑋} × 𝑓) ⊆ 𝑓
38	36, 37	sstri 3971	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓
39	38	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
40	35, 39	unssd 4166	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
41	34, 40	eqsstrid 4010	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42		trclimalb2 42153	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
43	16, 41, 42	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
44	33, 43	eqssd 3979	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45		sbcan 3809	. . . . . . . . . . . . 13 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})))
46		sbcan 3809	. . . . . . . . . . . . . . 15 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟))
47		fvex 6875	. . . . . . . . . . . . . . . . . 18 ⊢ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48		sbcssg 4501	. . . . . . . . . . . . . . . . . 18 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟))
49	47, 48	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟)
50		csbconstg 3892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 = 𝑅)
51	47, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 = 𝑅
52	47	csbvargi 4412	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
53	51, 52	sseq12i 3992	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
54	49, 53	bitri 274	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55		sbcssg 4501	. . . . . . . . . . . . . . . . . 18 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟))
56	47, 55	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟)
57		csbcog 6269	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ∘ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟))
58	47, 57	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ∘ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟)
59	52, 52	coeq12i 5839	. . . . . . . . . . . . . . . . . . 19 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ∘ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
60	58, 59	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
61	60, 52	sseq12i 3992	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
62	56, 61	bitri 274	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
63	54, 62	anbi12i 627	. . . . . . . . . . . . . . 15 ⊢ (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
64	46, 63	bitri 274	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
65		sbceq2g 4396	. . . . . . . . . . . . . . . 16 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋})))
66	47, 65	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}))
67		csbima12 6051	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
68	52	imaeq1i 6030	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
69		csbconstg 3892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋} = {𝑋})
70	47, 69	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋} = {𝑋}
71	70	imaeq2i 6031	. . . . . . . . . . . . . . . . 17 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
72	67, 68, 71	3eqtri 2763	. . . . . . . . . . . . . . . 16 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
73	72	eqeq2i 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
74	66, 73	bitri 274	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
75	64, 74	anbi12i 627	. . . . . . . . . . . . 13 ⊢ (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})) ↔ ((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})))
76	45, 75	sylbbr 235	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
77	19, 21, 44, 76	syl21anc 836	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
78	77	spesbcd 3857	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
79	78	ex 413	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
80		eqeq1 2735	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
81	80	rexbidv 3177	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
82		imaeq1 6028	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
83	82	eqeq2d 2742	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
84	83	rexab2 3675	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
85	81, 84	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
86	13, 85	elab 3648	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
87	79, 86	syl6ibr 251	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → 𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}))
88		intss1 4944	. . . . . . . 8 ⊢ (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓)
89	87, 88	syl6 35	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
90	89	alrimiv 1930	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
91		ssintab 4946	. . . . . 6 ⊢ (∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
92	90, 91	sylibr 233	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
93		ssintab 4946	. . . . . . 7 ⊢ (∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∀𝑔(∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔))
94	82	eqeq2d 2742	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
95	94	rexab2 3675	. . . . . . . . 9 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
96		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
97		imass1 6073	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
98	97	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99		imass1 6073	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
100		imaco 6223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
101		imass1 6073	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑟 ∘ 𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
102	100, 101	eqsstrrid 4011	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
103	99, 102	sylan9ss 3975	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104	98, 103	jca 512	. . . . . . . . . . . . 13 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
105	104	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106		vex 3463	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
107	106	imaex 7873	. . . . . . . . . . . . 13 ⊢ (𝑟 “ {𝑋}) ∈ V
108		imaundi 6122	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓))
109	108	sseq1i 3990	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
110		unss 4164	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
111	109, 110	bitr4i 277	. . . . . . . . . . . . . 14 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓))
112		imaeq2 6029	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (𝑅 “ 𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
113		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
114	112, 113	sseq12d 3995	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
115	114	cleq2lem 42035	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
116	111, 115	bitrid 282	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
117	107, 116	elab 3648	. . . . . . . . . . . 12 ⊢ ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
118	105, 117	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
119	96, 118	eqeltrd 2832	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
120	119	exlimiv 1933	. . . . . . . . 9 ⊢ (∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
121	95, 120	sylbi 216	. . . . . . . 8 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122		intss1 4944	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
123	121, 122	syl 17	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
124	93, 123	mpgbir 1801	. . . . . 6 ⊢ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}
125	124	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
126	92, 125	eqssd 3979	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
127	10, 126	eqtrd 2771	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
128	127	eleq2d 2818	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
129	8, 128	bitrd 278	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∃wrex 3069  Vcvv 3459  [wsbc 3757  ⦋csb 3873   ∪ cun 3926   ∩ cin 3927   ⊆ wss 3928  ∅c0 4302  {csn 4606  ⟨cop 4612  ∩ cint 4927   class class class wbr 5125   × cxp 5651  ran crn 5654   “ cima 5656   ∘ ccom 5657  ‘cfv 6516  t+ctcl 14897

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-n0 12438  df-z 12524  df-uz 12788  df-seq 13932  df-trcl 14899  df-relexp 14932

This theorem is referenced by:  dffrege76  42366