Theorem tfsconcatlem 43348

Description: Lemma for tfsconcatun 43349. (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 7713	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
2	1	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐵 ⊆ On)
3		oacl 8445	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4		eloni 6312	. . . . . . . . . . . . . . . 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 6312	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → Ord 𝐴)
7	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
8		ordeldif 43270	. . . . . . . . . . . . . . 15 ⊢ ((Ord (𝐴 +o 𝐵) ∧ Ord 𝐴) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶)))
9	5, 7, 8	syl2anc 584	. . . . . . . . . . . . . 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 1109	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
15		oawordex2 43338	. . . . . . . . 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 1136	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐴 ∈ On)
18		onss 7713	. . . . . . . . . . . . . . 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 4093	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ⊆ (On ∖ 𝐴))
21	20	sselda 3932	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
22	21	3impa 1109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
23		ordon 7705	. . . . . . . . . . . 12 ⊢ Ord On
24	17, 6	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → Ord 𝐴)
25		ordeldif 43270	. . . . . . . . . . . 12 ⊢ ((Ord On ∧ Ord 𝐴) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
26	23, 24, 25	sylancr 587	. . . . . . . . . . 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 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶))
30		oawordeu 8465	. . . . . . . . 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 4275	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
34		reurmo 3347	. . . . . . 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 3344	. . . . . 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 3664	. . . . . 6 ⊢ ∃*𝑥 𝑥 = (𝐹‘𝑦)
39	38	ax-gen 1796	. . . . 5 ⊢ ∀𝑦∃*𝑥 𝑥 = (𝐹‘𝑦)
40		moexexvw 2622	. . . . 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 586	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
42		df-rex 3055	. . . . . 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 1849	. . . . . 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 2542	. . . 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 6830	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
49	48	isseti 3452	. . . . . . . 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 3143	. . . . 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 3260	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
55		exmoeu 2575	. . . . 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 2737	. . . . 5 ⊢ ((𝐴 +o 𝑦) = 𝐶 ↔ 𝐶 = (𝐴 +o 𝑦))
60	59	anbi1i 624	. . . 4 ⊢ (((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
61	60	rexbii 3077	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
62	61	eubii 2579	. 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 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2110  ∃*wmo 2532  ∃!weu 2562  ∃wrex 3054  ∃!wreu 3342  ∃*wrmo 3343   ∖ cdif 3897   ⊆ wss 3900  Ord word 6301  Oncon0 6302  ‘cfv 6477  (class class class)co 7341   +_o coa 8377

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-oadd 8384

This theorem is referenced by:  tfsconcatun  43349  tfsconcatfn  43350  tfsconcatfv1  43351  tfsconcatfv2  43352