Theorem tfsconcatlem 43614

Description: Lemma for tfsconcatun 43615. (Contributed by RP, 23-Feb-2025.)

Assertion

Ref	Expression
tfsconcatlem	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem tfsconcatlem

Step	Hyp	Ref	Expression
1		onss 7732	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
2	1	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐵 ⊆ On)
3		oacl 8464	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4		eloni 6328	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
5	3, 4	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
6		eloni 6328	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → Ord 𝐴)
7	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
8		ordeldif 43536	. . . . . . . . . . . . . . 15 ⊢ ((Ord (𝐴 +o 𝐵) ∧ Ord 𝐴) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶)))
9	5, 7, 8	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶)))
10	9	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶))
11	10	ancomd 461	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵)))
12	11	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) → (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
13	12	imdistani 568	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
14	13	3impa 1110	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
15		oawordex2 43604	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
16	14, 15	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
17		simp1 1137	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐴 ∈ On)
18		onss 7732	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +o 𝐵) ∈ On → (𝐴 +o 𝐵) ⊆ On)
19	3, 18	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ On)
20	19	ssdifd 4098	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ⊆ (On ∖ 𝐴))
21	20	sselda 3934	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
22	21	3impa 1110	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
23		ordon 7724	. . . . . . . . . . . 12 ⊢ Ord On
24	17, 6	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → Ord 𝐴)
25		ordeldif 43536	. . . . . . . . . . . 12 ⊢ ((Ord On ∧ Ord 𝐴) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
26	23, 24, 25	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
27	22, 26	mpbid 232	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶))
28		anass 468	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶) ↔ (𝐴 ∈ On ∧ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
29	17, 27, 28	sylanbrc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶))
30		oawordeu 8484	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
32		reuss 4280	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ ∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 ∧ ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶) → ∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
33	2, 16, 31, 32	syl3anc 1374	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
34		reurmo 3354	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 → ∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
35	33, 34	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
36		df-rmo 3351	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
37	35, 36	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦(𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
38		moeq 3666	. . . . . 6 ⊢ ∃*𝑥 𝑥 = (𝐹‘𝑦)
39	38	ax-gen 1797	. . . . 5 ⊢ ∀𝑦∃*𝑥 𝑥 = (𝐹‘𝑦)
40		moexexvw 2629	. . . . 5 ⊢ ((∃*𝑦(𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ ∀𝑦∃*𝑥 𝑥 = (𝐹‘𝑦)) → ∃*𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
41	37, 39, 40	sylancl 587	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
42		df-rex 3062	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
43		anass 468	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
44	43	exbii 1850	. . . . . 6 ⊢ (∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
45	42, 44	bitr4i 278	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
46	45	mobii 2549	. . . 4 ⊢ (∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃*𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
47	41, 46	sylibr 234	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)))
48		fvex 6848	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
49	48	isseti 3459	. . . . . . . 8 ⊢ ∃𝑥 𝑥 = (𝐹‘𝑦)
50	49	jctr 524	. . . . . . 7 ⊢ ((𝐴 +o 𝑦) = 𝐶 → ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
51	50	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) ∧ 𝑦 ∈ 𝐵) → ((𝐴 +o 𝑦) = 𝐶 → ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦))))
52	51	reximdva 3150	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 → ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦))))
53	16, 52	mpd 15	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
54		rexcom4a 3267	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
55		exmoeu 2582	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ (∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) → ∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
56	54, 55	bitr3i 277	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)) ↔ (∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) → ∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
57	53, 56	sylib 218	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) → ∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
58	47, 57	mpd 15	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)))
59		eqcom 2744	. . . . 5 ⊢ ((𝐴 +o 𝑦) = 𝐶 ↔ 𝐶 = (𝐴 +o 𝑦))
60	59	anbi1i 625	. . . 4 ⊢ (((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
61	60	rexbii 3084	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
62	61	eubii 2586	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
63	58, 62	sylib 218	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569  ∃wrex 3061  ∃!wreu 3349  ∃*wrmo 3350   ∖ cdif 3899   ⊆ wss 3902  Ord word 6317  Oncon0 6318  ‘cfv 6493  (class class class)co 7360   +_o coa 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-oadd 8403

This theorem is referenced by:  tfsconcatun  43615  tfsconcatfn  43616  tfsconcatfv1  43617  tfsconcatfv2  43618