Theorem brtrclfv2 44080

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 5101	. . . 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 14935	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
4	3	breqd 5111	. . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
5	4	3ad2ant3 1136	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
6		elimasng 6056	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
7	6	3adant3 1133	. . 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 311	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋})))
9		intimasn 44010	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
10	9	3ad2ant1 1134	. . . 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 1195	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ∈ 𝑊)
12		snex 5385	. . . . . . . . . . . . . . 15 ⊢ {𝑋} ∈ V
13		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
14	12, 13	xpex 7708	. . . . . . . . . . . . . 14 ⊢ ({𝑋} × 𝑓) ∈ V
15		unexg 7698	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
16	11, 14, 15	sylancl 587	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17		trclfvlb 14943	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
18	17	unssad 4147	. . . . . . . . . . . . 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 14951	. . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ 𝑈)
23		snidg 4619	. . . . . . . . . . . . . . . . 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 4419	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
26	24, 24, 25	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27		xpima2 6150	. . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31		imass1 6068	. . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34		imaundir 6116	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36		imassrn 6038	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37		rnxpss 6138	. . . . . . . . . . . . . . . . . 18 ⊢ ran ({𝑋} × 𝑓) ⊆ 𝑓
38	36, 37	sstri 3945	. . . . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
41	34, 40	eqsstrid 3974	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42		trclimalb2 44079	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
43	16, 41, 42	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
44	33, 43	eqssd 3953	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45		sbcan 3792	. . . . . . . . . . . . 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 3792	. . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . 18 ⊢ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48		sbcssg 4476	. . . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . . . 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 4389	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
53	51, 52	sseq12i 3966	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
54	49, 53	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55		sbcssg 4476	. . . . . . . . . . . . . . . . . 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 6263	. . . . . . . . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
61	60, 52	sseq12i 3966	. . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . 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 4373	. . . . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
68	52	imaeq1i 6024	. . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . . . 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 6025	. . . . . . . . . . . . . . . . 17 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
72	67, 68, 71	3eqtri 2764	. . . . . . . . . . . . . . . 16 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
73	72	eqeq2i 2750	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
74	66, 73	bitri 275	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
75	64, 74	anbi12i 629	. . . . . . . . . . . . 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 236	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 3835	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
79	78	ex 412	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
80		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
81	80	rexbidv 3162	. . . . . . . . . . 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 6022	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
83	82	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
84	83	rexab2 3659	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
85	81, 84	bitrdi 287	. . . . . . . . . 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 3636	. . . . . . . . 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	imbitrrdi 252	. . . . . . . 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 4920	. . . . . . . 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 1929	. . . . . 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 4922	. . . . . 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 234	. . . . 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 4922	. . . . . . 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 2748	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
95	94	rexab2 3659	. . . . . . . . 9 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
96		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
97		imass1 6068	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
98	97	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99		imass1 6068	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
100		imaco 6217	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
101		imass1 6068	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑟 ∘ 𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
102	100, 101	eqsstrrid 3975	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
103	99, 102	sylan9ss 3949	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104	98, 103	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
105	104	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106		vex 3446	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
107	106	imaex 7866	. . . . . . . . . . . . 13 ⊢ (𝑟 “ {𝑋}) ∈ V
108		imaundi 6115	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓))
109	108	sseq1i 3964	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
110		unss 4144	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
111	109, 110	bitr4i 278	. . . . . . . . . . . . . 14 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓))
112		imaeq2 6023	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (𝑅 “ 𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
113		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
114	112, 113	sseq12d 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
115	114	cleq2lem 43961	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
116	111, 115	bitrid 283	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
117	107, 116	elab 3636	. . . . . . . . . . . 12 ⊢ ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
118	105, 117	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
119	96, 118	eqeltrd 2837	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
120	119	exlimiv 1932	. . . . . . . . 9 ⊢ (∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
121	95, 120	sylbi 217	. . . . . . . 8 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122		intss1 4920	. . . . . . . 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 3953	. . . 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 2772	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
128	127	eleq2d 2823	. 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 279	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∃wrex 3062  Vcvv 3442  [wsbc 3742  ⦋csb 3851   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  ∩ cint 4904   class class class wbr 5100   × cxp 5630  ran crn 5633   “ cima 5635   ∘ ccom 5636  ‘cfv 6500  t+ctcl 14920

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-seq 13937  df-trcl 14922  df-relexp 14955

This theorem is referenced by:  dffrege76  44292