brtrclfv2 - Mathbox for Richard Penner

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Theorem brtrclfv2 44486

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 5112	. . . 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 15056	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
4	3	breqd 5122	. . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
5	4	3ad2ant3 1153	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
6		elimasng 6093	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
7	6	3adant3 1150	. . 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 314	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋})))
9		intimasn 44416	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
10	9	3ad2ant1 1151	. . . 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 1212	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ∈ 𝑊)
12		snex 5412	. . . . . . . . . . . . . . 15 ⊢ {𝑋} ∈ V
13		vex 3461	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
14	12, 13	xpex 7754	. . . . . . . . . . . . . 14 ⊢ ({𝑋} × 𝑓) ∈ V
15		unexg 7747	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
16	11, 14, 15	sylancl 598	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17		trclfvlb 15064	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
18	17	unssad 4146	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
19	16, 18	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
20		trclfvcotrg 15072	. . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ 𝑈)
23		snidg 4628	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ {𝑋})
24	22, 23	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25		inelcm 4425	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
26	24, 24, 25	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27		xpima2 6184	. . . . . . . . . . . . . . 15 ⊢ (({𝑋} ∩ {𝑋}) ≠ ∅ → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
28	26, 27	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
29	16, 17	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
30	29	unssbd 4147	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31		imass1 6105	. . . . . . . . . . . . . . 15 ⊢ (({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
32	30, 31	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
33	28, 32	eqsstrrd 3973	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34		imaundir 6150	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36		imassrn 6075	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37		rnxpss 6172	. . . . . . . . . . . . . . . . . 18 ⊢ ran ({𝑋} × 𝑓) ⊆ 𝑓
38	36, 37	sstri 3947	. . . . . . . . . . . . . . . . 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 4145	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
41	34, 40	eqsstrid 3976	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42		trclimalb2 44485	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
43	16, 41, 42	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
44	33, 43	eqssd 3955	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45		sbcan 3795	. . . . . . . . . . . . 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 3795	. . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . . . 18 ⊢ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48		sbcssg 4484	. . . . . . . . . . . . . . . . . 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 3873	. . . . . . . . . . . . . . . . . . 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 4400	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
53	51, 52	sseq12i 3968	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
54	49, 53	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55		sbcssg 4484	. . . . . . . . . . . . . . . . . 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 6302	. . . . . . . . . . . . . . . . . . . 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 5851	. . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
61	60, 52	sseq12i 3968	. . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . 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 640	. . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . 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 4384	. . . . . . . . . . . . . . . 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 6083	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
68	52	imaeq1i 6061	. . . . . . . . . . . . . . . . 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 3873	. . . . . . . . . . . . . . . . . . 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 6062	. . . . . . . . . . . . . . . . 17 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
72	67, 68, 71	3eqtri 2792	. . . . . . . . . . . . . . . 16 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
73	72	eqeq2i 2778	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
74	66, 73	bitri 278	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
75	64, 74	anbi12i 640	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 851	. . . . . . . . . . 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 3837	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
79	78	ex 418	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
80		eqeq1 2769	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
81	80	rexbidv 3191	. . . . . . . . . . 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 6059	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
83	82	eqeq2d 2776	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
84	83	rexab2 3664	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
85	81, 84	bitrdi 290	. . . . . . . . . 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 3640	. . . . . . . . 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 255	. . . . . . . 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 4930	. . . . . . . 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 36	. . . . . . 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 1960	. . . . . 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 4932	. . . . . 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 237	. . . . 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 4932	. . . . . . 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 2776	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
95	94	rexab2 3664	. . . . . . . . 9 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
96		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
97		imass1 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
98	97	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99		imass1 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
100		imaco 6254	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
101		imass1 6105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑟 ∘ 𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
102	100, 101	eqsstrrid 3977	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
103	99, 102	sylan9ss 3951	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104	98, 103	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
105	104	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106		vex 3461	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
107	106	imaex 7913	. . . . . . . . . . . . 13 ⊢ (𝑟 “ {𝑋}) ∈ V
108		imaundi 6149	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓))
109	108	sseq1i 3966	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
110		unss 4143	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
111	109, 110	bitr4i 281	. . . . . . . . . . . . . 14 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓))
112		imaeq2 6060	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (𝑅 “ 𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
113		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
114	112, 113	sseq12d 3971	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
115	114	cleq2lem 44367	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
116	111, 115	bitrid 286	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
117	107, 116	elab 3640	. . . . . . . . . . . 12 ⊢ ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
118	105, 117	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
119	96, 118	eqeltrd 2865	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
120	119	exlimiv 1963	. . . . . . . . 9 ⊢ (∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
121	95, 120	sylbi 220	. . . . . . . 8 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122		intss1 4930	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
123	121, 122	syl 18	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
124	93, 123	mpgbir 1832	. . . . . 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 3955	. . . 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 2800	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
128	127	eleq2d 2851	. 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 282	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2146 {cab 2743 ≠ wne 2960 ∃wrex 3091 Vcvv 3457 [wsbc 3746 ⦋csb 3854 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 {csn 4591 ⟨cop 4597 ∩ cint 4914 class class class wbr 5111 × cxp 5661 ran crn 5664 “ cima 5666 ∘ ccom 5667 ‘cfv 6540 t+ctcl 15041

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-n0 12516 df-z 12603 df-uz 12874 df-seq 14051 df-trcl 15043 df-relexp 15076

This theorem is used by: dffrege76 44698

W3C validator