Theorem tfsconcatrev 44197

Description: If the domain of a transfinite sequence is an ordinal sum, the sequence can be decomposed into two sequences with domains corresponding to the addends. Theorem 2 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)

Hypothesis

Ref	Expression
tfsconcat.op	⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))

Assertion

Ref	Expression
tfsconcatrev	⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹 ↑m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))

Distinct variable groups:   𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧,𝐶   𝐷,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧   𝐹,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧   𝑢, + ,𝑣

Allowed substitution hints:   + (𝑥, 𝑦, 𝑧, 𝑎, 𝑏)

Proof of Theorem tfsconcatrev

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn3 6719	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
2	1	birani 509	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
3		fndm 6639	. . . . . . . 8 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → dom 𝐹 = (𝐶 +o 𝐷))
4	3	adantr 486	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 = (𝐶 +o 𝐷))
5		oacl 8526	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
6	5	adantl 487	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
7	4, 6	eqeltrd 2862	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 ∈ On)
8		fnfun 6636	. . . . . . 7 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → Fun 𝐹)
9	8	adantr 486	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → Fun 𝐹)
10		funrnex 7955	. . . . . 6 ⊢ (dom 𝐹 ∈ On → (Fun 𝐹 → ran 𝐹 ∈ V))
11	7, 9, 10	sylc 66	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐹 ∈ V)
12	11, 6	elmapd 8843	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹))
13	2, 12	mpbird 260	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)))
14		oaword1 8543	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
15	14	adantl 487	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ (𝐶 +o 𝐷))
16		elmapssres 8877	. . 3 ⊢ ((𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)) ∧ 𝐶 ⊆ (𝐶 +o 𝐷)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
17	13, 15, 16	syl2anc 596	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
18		simpl 488	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 Fn (𝐶 +o 𝐷))
19		oaordi 8537	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
20	19	ancoms 464	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
21	20	adantl 487	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
22	21	imp 412	. . . . 5 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
23		fnfvelrn 7077	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
24	18, 22, 23	syl2an2r 698	. . . 4 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
25	24	fmpttd 7112	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹)
26		simprr 785	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
27	11, 26	elmapd 8843	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹 ↑m 𝐷) ↔ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹))
28	25, 27	mpbird 260	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹 ↑m 𝐷))
29	18, 15	fnssresd 6660	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) Fn 𝐶)
30		fvex 6895	. . . . . . 7 ⊢ (𝐹‘(𝐶 +o 𝑑)) ∈ V
31		eqid 2762	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))
32	30, 31	fnmpti 6679	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷
33	32	a1i 11	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷)
34		simpr 490	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
35		tfsconcat.op	. . . . . 6 ⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))
36	35	tfsconcatun 44186	. . . . 5 ⊢ ((((𝐹 ↾ 𝐶) Fn 𝐶 ∧ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹 ↾ 𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}))
37	29, 33, 34, 36	syl21anc 851	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹 ↾ 𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}))
38		oveq2 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝐶 +o 𝑑) = (𝐶 +o 𝑧))
39	38	fveq2d 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝐹‘(𝐶 +o 𝑑)) = (𝐹‘(𝐶 +o 𝑧)))
40		fvex 6895	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(𝐶 +o 𝑧)) ∈ V
41	39, 31, 40	fvmpt 6990	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐷 → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
42	41	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
43		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐶 +o 𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐶 +o 𝑧)))
44	43	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝐹‘𝑥) = (𝐹‘(𝐶 +o 𝑧)))
45	42, 44	eqtr4d 2800	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘𝑥))
46	45	eqeq2d 2773	. . . . . . . . . . . . 13 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) ↔ 𝑦 = (𝐹‘𝑥)))
47	46	biimpd 232	. . . . . . . . . . . 12 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) → 𝑦 = (𝐹‘𝑥)))
48	47	expimpd 459	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧 ∈ 𝐷) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) → 𝑦 = (𝐹‘𝑥)))
49	48	rexlimdva 3165	. . . . . . . . . 10 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) → 𝑦 = (𝐹‘𝑥)))
50		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
51		eloni 6371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
52	5, 51	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
53		eloni 6371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ∈ On → Ord 𝐶)
54	53	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
55		ordeldif 44107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
56	52, 54, 55	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
57	56	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
58	57	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥))
59	58	ancomd 467	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷)))
60	50, 59	jca 521	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))))
61	60	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) → ((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))))
62		oawordex2 44175	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑧 ∈ 𝐷 (𝐶 +o 𝑧) = 𝑥)
63	61, 62	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) → ∃𝑧 ∈ 𝐷 (𝐶 +o 𝑧) = 𝑥)
64		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝐶 +o 𝑧) = 𝑥)
65	64	eqcomd 2768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑥 = (𝐶 +o 𝑧))
66	64	fveq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝐹‘(𝐶 +o 𝑧)) = (𝐹‘𝑥))
67	41	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
68		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑦 = (𝐹‘𝑥))
69	66, 67, 68	3eqtr4rd 2808	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))
70	65, 69	jca 521	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))
71	70	ex 418	. . . . . . . . . . . . 13 ⊢ (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) ∧ 𝑧 ∈ 𝐷) → ((𝐶 +o 𝑧) = 𝑥 → (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
72	71	reximdva 3177	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) → (∃𝑧 ∈ 𝐷 (𝐶 +o 𝑧) = 𝑥 → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
73	63, 72	mpd 16	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹‘𝑥)) → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))
74	73	ex 418	. . . . . . . . . 10 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹‘𝑥) → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
75	49, 74	impbid 215	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) ↔ 𝑦 = (𝐹‘𝑥)))
76		eldifi 4081	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) → 𝑥 ∈ (𝐶 +o 𝐷))
77		eqcom 2769	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
78		fnbrfvb 6932	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
79	77, 78	bitrid 286	. . . . . . . . . 10 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
80	18, 76, 79	syl2an 608	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
81	75, 80	bitrd 282	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) ↔ 𝑥𝐹𝑦))
82	81	pm5.32da 590	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))) ↔ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)))
83	82	opabbidv 5175	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)})
84		dfres2 6041	. . . . . 6 ⊢ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)}
85	83, 84	eqtr4di 2815	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))} = (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)))
86	85	uneq2d 4118	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}) = ((𝐹 ↾ 𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
87	37, 86	eqtrd 2797	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹 ↾ 𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
88		resundi 5990	. . . 4 ⊢ (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = ((𝐹 ↾ 𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)))
89	88	a1i 11	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = ((𝐹 ↾ 𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
90		undif 4441	. . . . . . 7 ⊢ (𝐶 ⊆ (𝐶 +o 𝐷) ↔ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
91	14, 90	sylib 221	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
92	91	adantl 487	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
93	92	reseq2d 5976	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = (𝐹 ↾ (𝐶 +o 𝐷)))
94		fnresdm 6655	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
95	94	adantr 486	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
96	93, 95	eqtrd 2797	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = 𝐹)
97	87, 89, 96	3eqtr2d 2803	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹)
98		dmres 6009	. . 3 ⊢ dom (𝐹 ↾ 𝐶) = (𝐶 ∩ dom 𝐹)
99	15, 4	sseqtrrd 3971	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ dom 𝐹)
100		dfss2 3920	. . . 4 ⊢ (𝐶 ⊆ dom 𝐹 ↔ (𝐶 ∩ dom 𝐹) = 𝐶)
101	99, 100	sylib 221	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∩ dom 𝐹) = 𝐶)
102	98, 101	eqtrid 2809	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝐹 ↾ 𝐶) = 𝐶)
103	30, 31	dmmpti 6680	. . 3 ⊢ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷
104	103	a1i 11	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)
105		oveq1 7424	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (𝑢 + 𝑣) = ((𝐹 ↾ 𝐶) + 𝑣))
106	105	eqeq1d 2764	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → ((𝑢 + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + 𝑣) = 𝐹))
107		dmeq 5891	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → dom 𝑢 = dom (𝐹 ↾ 𝐶))
108	107	eqeq1d 2764	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (dom 𝑢 = 𝐶 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
109	106, 108	3anbi12d 1465	. . 3 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷) ↔ (((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ∧ dom (𝐹 ↾ 𝐶) = 𝐶 ∧ dom 𝑣 = 𝐷)))
110		oveq2 7425	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((𝐹 ↾ 𝐶) + 𝑣) = ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))))
111	110	eqeq1d 2764	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹))
112		dmeq 5891	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → dom 𝑣 = dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))))
113	112	eqeq1d 2764	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (dom 𝑣 = 𝐷 ↔ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷))
114	111, 113	3anbi13d 1466	. . 3 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ∧ dom (𝐹 ↾ 𝐶) = 𝐶 ∧ dom 𝑣 = 𝐷) ↔ (((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹 ∧ dom (𝐹 ↾ 𝐶) = 𝐶 ∧ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)))
115	109, 114	rspc2ev 3592	. 2 ⊢ (((𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶) ∧ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹 ↑m 𝐷) ∧ (((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹 ∧ dom (𝐹 ↾ 𝐶) = 𝐶 ∧ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)) → ∃𝑢 ∈ (ran 𝐹 ↑m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
116	17, 28, 97, 102, 104, 115	syl113anc 1409	1 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹 ↑m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  dom cdm 5659  ran crn 5660   ↾ cres 5661  Ord word 6360  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   +_o coa 8456   ↑_m cmap 8830

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-pow 5334  ax-pr 5402  ax-un 7740

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 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-oadd 8463  df-map 8832

This theorem is used by: (None)