Theorem weiunlem2 36429

Description: Lemma for weiunpo 36431, weiunso 36432, weiunfr 36433, and weiunse 36434. (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)))}
weiunlem2.3	⊢ (𝜑 → 𝑅 We 𝐴)
weiunlem2.4	⊢ (𝜑 → 𝑅 Se 𝐴)

Assertion

Ref	Expression
weiunlem2	⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))

Distinct variable groups:   𝜑,𝑡   𝐴,𝑠,𝑡,𝑢,𝑣,𝑤,𝑥   𝑦,𝐴,𝑧,𝑥   𝐵,𝑠,𝑡,𝑢,𝑣,𝑤   𝑦,𝐵,𝑧   𝐹,𝑠,𝑡,𝑦,𝑧   𝑅,𝑠,𝑡,𝑢,𝑣,𝑤   𝑦,𝑅,𝑧   𝑆,𝑠,𝑡,𝑦,𝑧

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

Proof of Theorem weiunlem2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		weiunlem2.3	. 2 ⊢ (𝜑 → 𝑅 We 𝐴)
2		weiunlem2.4	. 2 ⊢ (𝜑 → 𝑅 Se 𝐴)
3		riotaex 7408	. . . . . 6 ⊢ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢) ∈ V
4		weiun.1	. . . . . 6 ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))
5	3, 4	fnmpti 6723	. . . . 5 ⊢ 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵
6	5	a1i 11	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
7		breq2 5170	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑟 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝑟))
8	7	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑟 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝑟))
9	8	ralbidv 3184	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑟 → (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢 ↔ ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟))
10	9	cbvriotavw 7414	. . . . . . . . . 10 ⊢ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟)
11		eleq1w 2827	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (𝑤 ∈ 𝐵 ↔ 𝑡 ∈ 𝐵))
12	11	rabbidv 3451	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
13		breq1 5169	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑠 → (𝑣𝑅𝑟 ↔ 𝑠𝑅𝑟))
14	13	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑠 → (¬ 𝑣𝑅𝑟 ↔ ¬ 𝑠𝑅𝑟))
15	14	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
16	12	raleqdv 3334	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑠𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
17	15, 16	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
18	12, 17	riotaeqbidv 7407	. . . . . . . . . 10 ⊢ (𝑤 = 𝑡 → (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
19	10, 18	eqtrid 2792	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
20	19, 4, 3	fvmpt3i 7034	. . . . . . . 8 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝐹‘𝑡) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
21	20	adantl 481	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐹‘𝑡) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
22		eliun 5019	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵)
23		rabn0 4412	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵)
24	22, 23	bitr4i 278	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅)
25		ssrab2 4103	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ⊆ 𝐴
26		wereu2 5697	. . . . . . . . . 10 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ ({𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅)) → ∃!𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
27	25, 26	mpanr1 702	. . . . . . . . 9 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅) → ∃!𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
28	24, 27	sylan2b 593	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∃!𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
29		riotacl2 7421	. . . . . . . 8 ⊢ (∃!𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟 → (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟})
30	28, 29	syl 17	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟})
31	21, 30	eqeltrd 2844	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟})
32		elrabi 3703	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟} → (𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
33		elrabi 3703	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → (𝐹‘𝑡) ∈ 𝐴)
34	31, 32, 33	3syl 18	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐹‘𝑡) ∈ 𝐴)
35	34	ralrimiva 3152	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝐹‘𝑡) ∈ 𝐴)
36		ffnfv 7153	. . . 4 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ↔ (𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝐹‘𝑡) ∈ 𝐴))
37	6, 35, 36	sylanbrc 582	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴)
38		dfsbcq 3806	. . . . . . 7 ⊢ (𝑠 = (𝐹‘𝑡) → ([𝑠 / 𝑥]𝑡 ∈ 𝐵 ↔ [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵))
39		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
40	39	elrabsf 3853	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ↔ (𝑠 ∈ 𝐴 ∧ [𝑠 / 𝑥]𝑡 ∈ 𝐵))
41	40	simprbi 496	. . . . . . 7 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → [𝑠 / 𝑥]𝑡 ∈ 𝐵)
42	38, 41	vtoclga 3589	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵)
43	31, 32, 42	3syl 18	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵)
44		sbcel2 4441	. . . . 5 ⊢ ([(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
45	43, 44	sylib 218	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
46	45	ralrimiva 3152	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
47		simpr 484	. . . . . . . . 9 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵))
48		sbcel2 4441	. . . . . . . . . . 11 ⊢ ([𝑠 / 𝑥]𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)
49	48	anbi2i 622	. . . . . . . . . 10 ⊢ ((𝑠 ∈ 𝐴 ∧ [𝑠 / 𝑥]𝑡 ∈ 𝐵) ↔ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵))
50	40, 49	bitri 275	. . . . . . . . 9 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ↔ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵))
51	47, 50	sylibr 234	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → 𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
52	51	ne0d 4365	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅)
53	52, 24	sylibr 234	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
54		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑟 = (𝐹‘𝑡) → (𝑠𝑅𝑟 ↔ 𝑠𝑅(𝐹‘𝑡)))
55	54	notbid 318	. . . . . . . . . 10 ⊢ (𝑟 = (𝐹‘𝑡) → (¬ 𝑠𝑅𝑟 ↔ ¬ 𝑠𝑅(𝐹‘𝑡)))
56	55	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑟 = (𝐹‘𝑡) → (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡)))
57	56	elrab 3708	. . . . . . . 8 ⊢ ((𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟} ↔ ((𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∧ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡)))
58	57	simprbi 496	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟} → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
59	31, 58	syl 17	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
60	53, 59	syldan 590	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
61		rsp 3253	. . . . 5 ⊢ (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡) → (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → ¬ 𝑠𝑅(𝐹‘𝑡)))
62	60, 51, 61	sylc 65	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → ¬ 𝑠𝑅(𝐹‘𝑡))
63	62	ralrimivva 3208	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡))
64	37, 46, 63	3jca 1128	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))
65	1, 2, 64	syl2anc 583	1 ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  {crab 3443  [wsbc 3804  ⦋csb 3921   ⊆ wss 3976  ∅c0 4352  ∪ ciun 5015   class class class wbr 5166  {copab 5228   ↦ cmpt 5249   Se wse 5650   We wwe 5651   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  ℩crio 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404

This theorem is referenced by:  weiunfrlem  36430  weiunpo  36431  weiunso  36432  weiunfr  36433  weiunse  36434