Theorem weiunlem 37221

Description: Lemma for weiunpo 37223, weiunso 37224, weiunfr 37225, and weiunse 37226. (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)

Assertion

Ref	Expression
weiunlem	⊢ (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 weiunlem

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		weiunlem.3	. 2 ⊢ (𝜑 → 𝑅 We 𝐴)
2		weiunlem.4	. 2 ⊢ (𝜑 → 𝑅 Se 𝐴)
3		riotaex 7373	. . . . . 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 6674	. . . . 5 ⊢ 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵
6	5	a1i 11	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
7		breq2 5107	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑟 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝑟))
8	7	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑟 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝑟))
9	8	ralbidv 3186	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑟 → (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢 ↔ ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟))
10	9	cbvriotavw 7379	. . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (𝑤 ∈ 𝐵 ↔ 𝑡 ∈ 𝐵))
12	11	rabbidv 3420	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
13		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑠 → (𝑣𝑅𝑟 ↔ 𝑠𝑅𝑟))
14	13	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑠 → (¬ 𝑣𝑅𝑟 ↔ ¬ 𝑠𝑅𝑟))
15	14	cbvralvw 3241	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
16	12	raleqdv 3320	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑠𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
17	15, 16	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
18	12, 17	riotaeqbidv 7372	. . . . . . . . . 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 2808	. . . . . . . . 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 6991	. . . . . . . 8 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝐹‘𝑡) = (℩𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟))
21	20	adantl 487	. . . . . . 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 4955	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵)
23		rabn0 4339	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵)
24	22, 23	bitr4i 281	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅)
25		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ⊆ 𝐴
26		wereu2 5648	. . . . . . . . . 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 716	. . . . . . . . 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 606	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∃!𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵}∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟)
29		riotacl2 7385	. . . . . . . 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 18	. . . . . . 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 2861	. . . . . 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 3641	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟} → (𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
33		elrabi 3641	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → (𝐹‘𝑡) ∈ 𝐴)
34	31, 32, 33	3syl 19	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐹‘𝑡) ∈ 𝐴)
35	34	ralrimiva 3155	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝐹‘𝑡) ∈ 𝐴)
36		ffnfv 7111	. . . 4 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ↔ (𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝐹‘𝑡) ∈ 𝐴))
37	6, 35, 36	sylanbrc 595	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴)
38		dfsbcq 3741	. . . . . . 7 ⊢ (𝑠 = (𝐹‘𝑡) → ([𝑠 / 𝑥]𝑡 ∈ 𝐵 ↔ [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵))
39		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
40	39	elrabsf 3784	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ↔ (𝑠 ∈ 𝐴 ∧ [𝑠 / 𝑥]𝑡 ∈ 𝐵))
41	40	simprbi 503	. . . . . . 7 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → [𝑠 / 𝑥]𝑡 ∈ 𝐵)
42	38, 41	vtoclga 3537	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵)
43	31, 32, 42	3syl 19	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → [(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵)
44		sbcel2 4376	. . . . 5 ⊢ ([(𝐹‘𝑡) / 𝑥]𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
45	43, 44	sylib 221	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
46	45	ralrimiva 3155	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵)
47		sbcel2 4376	. . . . . . . . . . 11 ⊢ ([𝑠 / 𝑥]𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)
48	47	anbi2i 635	. . . . . . . . . 10 ⊢ ((𝑠 ∈ 𝐴 ∧ [𝑠 / 𝑥]𝑡 ∈ 𝐵) ↔ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵))
49	40, 48	bitri 278	. . . . . . . . 9 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ↔ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵))
50	49	bilanri 512	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → 𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵})
51	50	ne0d 4288	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ≠ ∅)
52	51, 24	sylibr 237	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
53		breq2 5107	. . . . . . . . . . 11 ⊢ (𝑟 = (𝐹‘𝑡) → (𝑠𝑅𝑟 ↔ 𝑠𝑅(𝐹‘𝑡)))
54	53	notbid 321	. . . . . . . . . 10 ⊢ (𝑟 = (𝐹‘𝑡) → (¬ 𝑠𝑅𝑟 ↔ ¬ 𝑠𝑅(𝐹‘𝑡)))
55	54	ralbidv 3186	. . . . . . . . 9 ⊢ (𝑟 = (𝐹‘𝑡) → (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟 ↔ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡)))
56	55	elrab 3645	. . . . . . . 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)))
57	56	simprbi 503	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ {𝑟 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ∣ ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅𝑟} → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
58	31, 57	syl 18	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
59	52, 58	syldan 603	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → ∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡))
60		rsp 3251	. . . . 5 ⊢ (∀𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} ¬ 𝑠𝑅(𝐹‘𝑡) → (𝑠 ∈ {𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵} → ¬ 𝑠𝑅(𝐹‘𝑡)))
61	59, 50, 60	sylc 66	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵)) → ¬ 𝑠𝑅(𝐹‘𝑡))
62	61	ralrimivva 3206	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡))
63	37, 46, 62	3jca 1146	. 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)))
64	1, 2, 63	syl2anc 596	1 ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364  {crab 3413  [wsbc 3739  ⦋csb 3847   ⊆ wss 3899  ∅c0 4279  ∪ ciun 4951   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   Se wse 5602   We wwe 5603   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℩crio 7368

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-riota 7369

This theorem is used by:  weiunfrlem  37222  weiunpo  37223  weiunso  37224  weiunfr  37225  weiunse  37226