Theorem weiunfrlem 36658

Description: Lemma for weiunfr 36661. (Contributed by Matthew House, 23-Aug-2025.)

Hypotheses

Ref	Expression
weiun.1	⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))
weiun.2	⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}
weiunlem.3	⊢ (𝜑 → 𝑅 We 𝐴)
weiunlem.4	⊢ (𝜑 → 𝑅 Se 𝐴)
weiunfrlem.5	⊢ 𝐸 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)
weiunfrlem.6	⊢ (𝜑 → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
weiunfrlem.7	⊢ (𝜑 → 𝑟 ≠ ∅)

Assertion

Ref	Expression
weiunfrlem	⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸))

Distinct variable groups:   𝜑,𝑡   𝐴,𝑝,𝑞,𝑟,𝑡,𝑢,𝑣,𝑤,𝑥   𝑦,𝐴,𝑧,𝑥   𝐵,𝑝,𝑞,𝑟,𝑡,𝑢,𝑣,𝑤   𝑦,𝐵,𝑧   𝑡,𝐸   𝐹,𝑝,𝑞,𝑟,𝑡,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑡,𝑢,𝑣,𝑤   𝑦,𝑅,𝑧   𝑆,𝑝,𝑞,𝑟,𝑡,𝑦,𝑧   𝑇,𝑝,𝑞,𝑟

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑟,𝑞,𝑝)   𝐵(𝑥)   𝑅(𝑥)   𝑆(𝑥,𝑤,𝑣,𝑢)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡)   𝐸(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑟,𝑞,𝑝)   𝐹(𝑥,𝑤,𝑣,𝑢)

Proof of Theorem weiunfrlem

Dummy variables 𝑛 𝑜 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		weiunlem.3	. . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐴)
2		weiunlem.4	. . . . . . 7 ⊢ (𝜑 → 𝑅 Se 𝐴)
3		weiun.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))
4		weiun.2	. . . . . . . . . 10 ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}
5	3, 4, 1, 2	weiunlem 36657	. . . . . . . . 9 ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))
6	5	simp1d 1142	. . . . . . . 8 ⊢ (𝜑 → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴)
7	6	fimassd 6683	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝑟) ⊆ 𝐴)
8		weiunfrlem.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9	6	fdmd 6672	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = ∪ 𝑥 ∈ 𝐴 𝐵)
10	8, 9	sseqtrrd 3971	. . . . . . . . . 10 ⊢ (𝜑 → 𝑟 ⊆ dom 𝐹)
11		sseqin2 4175	. . . . . . . . . 10 ⊢ (𝑟 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑟) = 𝑟)
12	10, 11	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝐹 ∩ 𝑟) = 𝑟)
13		weiunfrlem.7	. . . . . . . . 9 ⊢ (𝜑 → 𝑟 ≠ ∅)
14	12, 13	eqnetrd 2999	. . . . . . . 8 ⊢ (𝜑 → (dom 𝐹 ∩ 𝑟) ≠ ∅)
15	14	imadisjlnd 6040	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝑟) ≠ ∅)
16		wereu2 5621	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐹 “ 𝑟) ⊆ 𝐴 ∧ (𝐹 “ 𝑟) ≠ ∅)) → ∃!𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)
17	1, 2, 7, 15, 16	syl22anc 838	. . . . . 6 ⊢ (𝜑 → ∃!𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)
18		riotacl2 7331	. . . . . 6 ⊢ (∃!𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝 → (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ {𝑝 ∈ (𝐹 “ 𝑟) ∣ ∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝})
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ {𝑝 ∈ (𝐹 “ 𝑟) ∣ ∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝})
20		weiunfrlem.5	. . . . 5 ⊢ 𝐸 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)
21		simpr 484	. . . . . . . . 9 ⊢ ((𝑛 = 𝑝 ∧ 𝑜 = 𝑞) → 𝑜 = 𝑞)
22		simpl 482	. . . . . . . . 9 ⊢ ((𝑛 = 𝑝 ∧ 𝑜 = 𝑞) → 𝑛 = 𝑝)
23	21, 22	breq12d 5111	. . . . . . . 8 ⊢ ((𝑛 = 𝑝 ∧ 𝑜 = 𝑞) → (𝑜𝑅𝑛 ↔ 𝑞𝑅𝑝))
24	23	notbid 318	. . . . . . 7 ⊢ ((𝑛 = 𝑝 ∧ 𝑜 = 𝑞) → (¬ 𝑜𝑅𝑛 ↔ ¬ 𝑞𝑅𝑝))
25	24	cbvraldva 3216	. . . . . 6 ⊢ (𝑛 = 𝑝 → (∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝑛 ↔ ∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))
26	25	cbvrabv 3409	. . . . 5 ⊢ {𝑛 ∈ (𝐹 “ 𝑟) ∣ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝑛} = {𝑝 ∈ (𝐹 “ 𝑟) ∣ ∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝}
27	19, 20, 26	3eltr4g 2853	. . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑛 ∈ (𝐹 “ 𝑟) ∣ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝑛})
28		breq2 5102	. . . . . . 7 ⊢ (𝑛 = 𝐸 → (𝑜𝑅𝑛 ↔ 𝑜𝑅𝐸))
29	28	notbid 318	. . . . . 6 ⊢ (𝑛 = 𝐸 → (¬ 𝑜𝑅𝑛 ↔ ¬ 𝑜𝑅𝐸))
30	29	ralbidv 3159	. . . . 5 ⊢ (𝑛 = 𝐸 → (∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝑛 ↔ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸))
31	30	elrab 3646	. . . 4 ⊢ (𝐸 ∈ {𝑛 ∈ (𝐹 “ 𝑟) ∣ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝑛} ↔ (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸))
32	27, 31	sylib 218	. . 3 ⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸))
33	32	simpld 494	. 2 ⊢ (𝜑 → 𝐸 ∈ (𝐹 “ 𝑟))
34	32	simprd 495	. . 3 ⊢ (𝜑 → ∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸)
35	6	ffnd 6663	. . . 4 ⊢ (𝜑 → 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
36		breq1 5101	. . . . . 6 ⊢ (𝑜 = (𝐹‘𝑡) → (𝑜𝑅𝐸 ↔ (𝐹‘𝑡)𝑅𝐸))
37	36	notbid 318	. . . . 5 ⊢ (𝑜 = (𝐹‘𝑡) → (¬ 𝑜𝑅𝐸 ↔ ¬ (𝐹‘𝑡)𝑅𝐸))
38	37	ralima 7183	. . . 4 ⊢ ((𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸 ↔ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸))
39	35, 8, 38	syl2anc 584	. . 3 ⊢ (𝜑 → (∀𝑜 ∈ (𝐹 “ 𝑟) ¬ 𝑜𝑅𝐸 ↔ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸))
40	34, 39	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸)
41		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵))
42	41	elin1d 4156	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑡 ∈ 𝑟)
43		rspa 3225	. . . . 5 ⊢ ((∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ 𝑡 ∈ 𝑟) → ¬ (𝐹‘𝑡)𝑅𝐸)
44	40, 42, 43	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → ¬ (𝐹‘𝑡)𝑅𝐸)
45		csbeq1 3852	. . . . . . 7 ⊢ (𝑠 = 𝐸 → ⦋𝑠 / 𝑥⦌𝐵 = ⦋𝐸 / 𝑥⦌𝐵)
46		breq1 5101	. . . . . . . 8 ⊢ (𝑠 = 𝐸 → (𝑠𝑅(𝐹‘𝑡) ↔ 𝐸𝑅(𝐹‘𝑡)))
47	46	notbid 318	. . . . . . 7 ⊢ (𝑠 = 𝐸 → (¬ 𝑠𝑅(𝐹‘𝑡) ↔ ¬ 𝐸𝑅(𝐹‘𝑡)))
48	45, 47	raleqbidv 3316	. . . . . 6 ⊢ (𝑠 = 𝐸 → (∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡) ↔ ∀𝑡 ∈ ⦋ 𝐸 / 𝑥⦌𝐵 ¬ 𝐸𝑅(𝐹‘𝑡)))
49	5	simp3d 1144	. . . . . 6 ⊢ (𝜑 → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡))
50	7, 33	sseldd 3934	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
51	48, 49, 50	rspcdva 3577	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ ⦋ 𝐸 / 𝑥⦌𝐵 ¬ 𝐸𝑅(𝐹‘𝑡))
52	41	elin2d 4157	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑡 ∈ ⦋𝐸 / 𝑥⦌𝐵)
53		rspa 3225	. . . . 5 ⊢ ((∀𝑡 ∈ ⦋ 𝐸 / 𝑥⦌𝐵 ¬ 𝐸𝑅(𝐹‘𝑡) ∧ 𝑡 ∈ ⦋𝐸 / 𝑥⦌𝐵) → ¬ 𝐸𝑅(𝐹‘𝑡))
54	51, 52, 53	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → ¬ 𝐸𝑅(𝐹‘𝑡))
55		weso 5615	. . . . . . 7 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
56	1, 55	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑅 Or 𝐴)
58	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴)
59	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
60	59, 42	sseldd 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
61	58, 60	ffvelcdmd 7030	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → (𝐹‘𝑡) ∈ 𝐴)
62	50	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → 𝐸 ∈ 𝐴)
63		sotrieq2 5564	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ ((𝐹‘𝑡) ∈ 𝐴 ∧ 𝐸 ∈ 𝐴)) → ((𝐹‘𝑡) = 𝐸 ↔ (¬ (𝐹‘𝑡)𝑅𝐸 ∧ ¬ 𝐸𝑅(𝐹‘𝑡))))
64	57, 61, 62, 63	syl12anc 836	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → ((𝐹‘𝑡) = 𝐸 ↔ (¬ (𝐹‘𝑡)𝑅𝐸 ∧ ¬ 𝐸𝑅(𝐹‘𝑡))))
65	44, 54, 64	mpbir2and 713	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)) → (𝐹‘𝑡) = 𝐸)
66	65	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸)
67	33, 40, 66	3jca 1128	1 ⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃!wreu 3348  {crab 3399  ⦋csb 3849   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ∪ ciun 4946   class class class wbr 5098  {copab 5160   ↦ cmpt 5179   Or wor 5531   Se wse 5575   We wwe 5576  dom cdm 5624   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  ℩crio 7314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7315

This theorem is referenced by:  weiunfr  36661