Theorem tfsconcatlem 44164

Description: Lemma for tfsconcatun 44165. (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 7787	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
2	1	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐵 ⊆ On)
3		oacl 8525	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4		eloni 6371	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
5	3, 4	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
6		eloni 6371	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → Ord 𝐴)
7	6	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
8		ordeldif 44086	. . . . . . . . . . . . . . 15 ⊢ ((Ord (𝐴 +o 𝐵) ∧ Ord 𝐴) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶)))
9	5, 7, 8	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶)))
10	9	biimpa 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝐶))
11	10	ancomd 467	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵)))
12	11	ex 418	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) → (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
13	12	imdistani 579	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
14	13	3impa 1127	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))))
15		oawordex2 44154	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
16	14, 15	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
17		simp1 1154	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐴 ∈ On)
18		onss 7787	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +o 𝐵) ∈ On → (𝐴 +o 𝐵) ⊆ On)
19	3, 18	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ On)
20	19	ssdifd 4095	. . . . . . . . . . . . 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 1127	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
23		ordon 7779	. . . . . . . . . . . 12 ⊢ Ord On
24	17, 6	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → Ord 𝐴)
25		ordeldif 44086	. . . . . . . . . . . 12 ⊢ ((Ord On ∧ Ord 𝐴) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
26	23, 24, 25	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
27	22, 26	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶))
28		anass 474	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶) ↔ (𝐴 ∈ On ∧ (𝐶 ∈ On ∧ 𝐴 ⊆ 𝐶)))
29	17, 27, 28	sylanbrc 595	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶))
30		oawordeu 8545	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐶) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
31	29, 30	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
32		reuss 4276	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
34		reurmo 3370	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 → ∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
35	33, 34	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶)
36		df-rmo 3367	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐵 (𝐴 +o 𝑦) = 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
37	35, 36	sylib 221	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦(𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
38		moeq 3668	. . . . . 6 ⊢ ∃*𝑥 𝑥 = (𝐹‘𝑦)
39	38	ax-gen 1828	. . . . 5 ⊢ ∀𝑦∃*𝑥 𝑥 = (𝐹‘𝑦)
40		moexexvw 2655	. . . . 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 598	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
42		df-rex 3089	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
43		anass 474	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦))))
44	43	exbii 1881	. . . . . 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 281	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦((𝑦 ∈ 𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹‘𝑦)))
46	45	mobii 2575	. . . 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 237	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)))
48		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
49	48	isseti 3471	. . . . . . . 8 ⊢ ∃𝑥 𝑥 = (𝐹‘𝑦)
50	49	jctr 534	. . . . . . 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 3177	. . . . 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 16	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
54		rexcom4a 3294	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹‘𝑦)))
55		exmoeu 2608	. . . . 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 280	. . . 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 221	. . 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 16	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)))
59		eqcom 2769	. . . . 5 ⊢ ((𝐴 +o 𝑦) = 𝐶 ↔ 𝐶 = (𝐴 +o 𝑦))
60	59	anbi1i 636	. . . 4 ⊢ (((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
61	60	rexbii 3111	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ 𝑥 = (𝐹‘𝑦)) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
62	61	eubii 2612	. 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 221	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2564  ∃!weu 2595  ∃wrex 3088  ∃!wreu 3365  ∃*wrmo 3366   ∖ cdif 3899   ⊆ wss 3902  Ord word 6360  Oncon0 6361  ‘cfv 6537  (class class class)co 7416   +_o coa 8455

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-oadd 8462

This theorem is used by:  tfsconcatun  44165  tfsconcatfn  44166  tfsconcatfv1  44167  tfsconcatfv2  44168