Theorem brtrclfv2 37038

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 4584	. . . 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 13589	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
4	3	breqd 4594	. . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
5	4	3ad2ant3 1077	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑋∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑌))
6		elimasng 5410	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}))
7	6	3adant3 1074	. . 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 299	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋})))
9		intimasn 36968	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
10	9	3ad2ant1 1075	. . . 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 1059	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ∈ 𝑊)
12		snex 4835	. . . . . . . . . . . . . . 15 ⊢ {𝑋} ∈ V
13		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
14	12, 13	xpex 6860	. . . . . . . . . . . . . 14 ⊢ ({𝑋} × 𝑓) ∈ V
15		unexg 6857	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
16	11, 14, 15	sylancl 693	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17		trclfvlb 13597	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
18	17	unssad 3752	. . . . . . . . . . . . 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 13605	. . . . . . . . . . . . 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 1057	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ 𝑈)
23		snidg 4153	. . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
26	24, 24, 25	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27		xpima2 5497	. . . . . . . . . . . . . . 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 3753	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31		imass1 5419	. . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34		imaundir 5465	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36		imassrn 5396	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37		rnxpss 5485	. . . . . . . . . . . . . . . . . 18 ⊢ ran ({𝑋} × 𝑓) ⊆ 𝑓
38	36, 37	sstri 3577	. . . . . . . . . . . . . . . . 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 3751	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
41	34, 40	syl5eqss 3612	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42		trclimalb2 37037	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
43	16, 41, 42	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
44	33, 43	eqssd 3585	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45		sbcan 3445	. . . . . . . . . . . . 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 3445	. . . . . . . . . . . . . . 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 6113	. . . . . . . . . . . . . . . . . 18 ⊢ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48		sbcssg 4035	. . . . . . . . . . . . . . . . . 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 3512	. . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑅 ⊆ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55	49, 54	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅 ⊆ 𝑟 ↔ 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
56		sbcssg 4035	. . . . . . . . . . . . . . . . . 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 36960	. . . . . . . . . . . . . . . . . . . 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 5207	. . . . . . . . . . . . . . . . . . 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 2632	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 ∘ 𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
62	61, 53	sseq12i 3594	. . . . . . . . . . . . . . . . 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 263	. . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . 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 263	. . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . 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 5402	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = (⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌𝑟 “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋})
69	53	imaeq1i 5382	. . . . . . . . . . . . . . . . 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 3512	. . . . . . . . . . . . . . . . . . 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 5383	. . . . . . . . . . . . . . . . 17 ⊢ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
73	68, 69, 72	3eqtri 2636	. . . . . . . . . . . . . . . 16 ⊢ ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
74	73	eqeq2i 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⦋(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟⦌(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
75	67, 74	bitri 263	. . . . . . . . . . . . . 14 ⊢ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
76	65, 75	anbi12i 729	. . . . . . . . . . . . 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 225	. . . . . . . . . . . 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 1317	. . . . . . . . . . 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 3488	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
80	79	ex 449	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
81		eqeq1 2614	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
82	81	rexbidv 3034	. . . . . . . . . . 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 5380	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
84	83	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
85	84	rexab2 3340	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
86	82, 85	syl6bb 275	. . . . . . . . . 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 3319	. . . . . . . . 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 241	. . . . . . . 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 4427	. . . . . . . 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 34	. . . . . . 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 1842	. . . . . 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 4429	. . . . . 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 223	. . . . 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 4429	. . . . . . 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 2620	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
96	95	rexab2 3340	. . . . . . . . 9 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
97		simpr 476	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
98		imass1 5419	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99	98	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
100		imass1 5419	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ⊆ 𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
101		imaco 5557	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
102		imass1 5419	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑟 ∘ 𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
103	101, 102	syl5eqssr 3613	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104	100, 103	sylan9ss 3581	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
105	99, 104	jca 553	. . . . . . . . . . . . 13 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106	105	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
107		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
108		imaexg 6995	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ V → (𝑟 “ {𝑋}) ∈ V)
109	107, 108	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑟 “ {𝑋}) ∈ V
110		imaundi 5464	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓))
111	110	sseq1i 3592	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
112		unss 3749	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅 “ 𝑓)) ⊆ 𝑓)
113	111, 112	bitr4i 266	. . . . . . . . . . . . . 14 ⊢ ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓))
114		imaeq2 5381	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (𝑅 “ 𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
115		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
116	114, 115	sseq12d 3597	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
117	116	cleq2lem 36933	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅 “ 𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
118	113, 117	syl5bb 271	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
119	109, 118	elab 3319	. . . . . . . . . . . 12 ⊢ ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
120	106, 119	sylibr 223	. . . . . . . . . . 11 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
121	97, 120	eqeltrd 2688	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122	121	exlimiv 1845	. . . . . . . . 9 ⊢ (∃𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
123	96, 122	sylbi 206	. . . . . . . 8 ⊢ (∃𝑠 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
124		intss1 4427	. . . . . . . 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 1717	. . . . . 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 3585	. . . 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 2644	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} “ {𝑋}) = ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
130	129	eleq2d 2673	. 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 267	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∩ cint 4410   class class class wbr 4583   × cxp 5036  ran crn 5039   “ cima 5041   ∘ ccom 5042  ‘cfv 5804  t+ctcl 13572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-trcl 13574  df-relexp 13609

This theorem is referenced by:  dffrege76  37253