Theorem brtrclfv2 38019

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 4654	. . . 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 13741	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
4	3	breqd 4664	. . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
5	4	3ad2ant3 1084	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
6		elimasng 5491	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
7	6	3adant3 1081	. . 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 300	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋})))
9		intimasn 37949	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
10	9	3ad2ant1 1082	. . . 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 1066	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ∈ 𝑊)
12		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {𝑋} ∈ V
13		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
14	12, 13	xpex 6962	. . . . . . . . . . . . . 14 ⊢ ({𝑋} × 𝑓) ∈ V
15		unexg 6959	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
16	11, 14, 15	sylancl 694	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17		trclfvlb 13749	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
18	17	unssad 3790	. . . . . . . . . . . . 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 13757	. . . . . . . . . . . . 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 1064	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ 𝑈)
23		snidg 4206	. . . . . . . . . . . . . . . . 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 4032	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
26	24, 24, 25	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27		xpima2 5578	. . . . . . . . . . . . . . 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 3791	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31		imass1 5500	. . . . . . . . . . . . . . 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	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34		imaundir 5546	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36		imassrn 5477	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37		rnxpss 5566	. . . . . . . . . . . . . . . . . 18 ⊢ ran ({𝑋} × 𝑓) ⊆ 𝑓
38	36, 37	sstri 3612	. . . . . . . . . . . . . . . . 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 3789	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
41	34, 40	syl5eqss 3649	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42		trclimalb2 38018	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
43	16, 41, 42	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
44	33, 43	eqssd 3620	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45		sbcan 3478	. . . . . . . . . . . . 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 3478	. . . . . . . . . . . . . . 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 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48		sbcssg 4085	. . . . . . . . . . . . . . . . . 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 3546	. . . . . . . . . . . . . . . . . . 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		csbvarg 4003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
53	47, 52	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
54	51, 53	sseq12i 3631	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55	49, 54	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
56		sbcssg 4085	. . . . . . . . . . . . . . . . . 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))
57	47, 56	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)
58		csbcog 37941	. . . . . . . . . . . . . . . . . . . 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))
59	47, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ∘ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟)
60	53, 53	coeq12i 5285	. . . . . . . . . . . . . . . . . . 19 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ∘ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
61	59, 60	eqtri 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
62	61, 53	sseq12i 3631	. . . . . . . . . . . . . . . . 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))))
63	57, 62	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
64	55, 63	anbi12i 733	. . . . . . . . . . . . . . 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)))))
65	46, 64	bitri 264	. . . . . . . . . . . . . 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)))))
66		sbceq2g 3990	. . . . . . . . . . . . . . . 16 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋})))
67	47, 66	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}))
68		csbima12 5483	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
69	53	imaeq1i 5463	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
70		csbconstg 3546	. . . . . . . . . . . . . . . . . . 19 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋} = {𝑋})
71	47, 70	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋} = {𝑋}
72	71	imaeq2i 5464	. . . . . . . . . . . . . . . . 17 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
73	68, 69, 72	3eqtri 2648	. . . . . . . . . . . . . . . 16 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
74	73	eqeq2i 2634	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
75	67, 74	bitri 264	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
76	65, 75	anbi12i 733	. . . . . . . . . . . . 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})))
77	45, 76	sylbbr 226	. . . . . . . . . . . 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})))
78	19, 21, 44, 77	syl21anc 1325	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
79	78	spesbcd 3522	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
80	79	ex 450	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
81		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
82	81	rexbidv 3052	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
83		imaeq1 5461	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
84	83	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
85	84	rexab2 3373	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
86	82, 85	syl6bb 276	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
87	13, 86	elab 3350	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
88	80, 87	syl6ibr 242	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → 𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}))
89		intss1 4492	. . . . . . . 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)
90	88, 89	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))
91	90	alrimiv 1855	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
92		ssintab 4494	. . . . . 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))
93	91, 92	sylibr 224	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
94		ssintab 4494	. . . . . . 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))
95	83	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
96	95	rexab2 3373	. . . . . . . . 9 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
97		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
98		imass1 5500	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99	98	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
100		imass1 5500	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
101		imaco 5640	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
102		imass1 5500	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑟 ∘ 𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
103	101, 102	syl5eqssr 3650	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104	100, 103	sylan9ss 3616	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
105	99, 104	jca 554	. . . . . . . . . . . . 13 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106	105	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
107		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
108		imaexg 7103	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ V → (𝑟 “ {𝑋}) ∈ V)
109	107, 108	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑟 “ {𝑋}) ∈ V
110		imaundi 5545	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓))
111	110	sseq1i 3629	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
112		unss 3787	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
113	111, 112	bitr4i 267	. . . . . . . . . . . . . 14 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓))
114		imaeq2 5462	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (𝑅 “ 𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
115		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
116	114, 115	sseq12d 3634	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
117	116	cleq2lem 37914	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
118	113, 117	syl5bb 272	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
119	109, 118	elab 3350	. . . . . . . . . . . 12 ⊢ ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
120	106, 119	sylibr 224	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
121	97, 120	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122	121	exlimiv 1858	. . . . . . . . 9 ⊢ (∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
123	96, 122	sylbi 207	. . . . . . . 8 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
124		intss1 4492	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
125	123, 124	syl 17	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
126	94, 125	mpgbir 1726	. . . . . 6 ⊢ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}
127	126	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})})
128	93, 127	eqssd 3620	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
129	10, 128	eqtrd 2656	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
130	129	eleq2d 2687	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
131	8, 130	bitrd 268	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653   × cxp 5112  ran crn 5115   “ cima 5117   ∘ ccom 5118  ‘cfv 5888  t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-trcl 13726  df-relexp 13761

This theorem is referenced by:  dffrege76  38233