Theorem tfsconcatrev 43873

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 6693	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
2	1	birani 506	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
3		fndm 6613	. . . . . . . 8 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → dom 𝐹 = (𝐶 +o 𝐷))
4	3	adantr 483	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 = (𝐶 +o 𝐷))
5		oacl 8492	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
6	5	adantl 484	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
7	4, 6	eqeltrd 2856	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 ∈ On)
8		fnfun 6610	. . . . . . 7 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → Fun 𝐹)
9	8	adantr 483	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → Fun 𝐹)
10		funrnex 7924	. . . . . 6 ⊢ (dom 𝐹 ∈ On → (Fun 𝐹 → ran 𝐹 ∈ V))
11	7, 9, 10	sylc 65	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐹 ∈ V)
12	11, 6	elmapd 8810	. . . 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 259	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)))
14		oaword1 8509	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
15	14	adantl 484	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ (𝐶 +o 𝐷))
16		elmapssres 8837	. . 3 ⊢ ((𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)) ∧ 𝐶 ⊆ (𝐶 +o 𝐷)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
17	13, 15, 16	syl2anc 592	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
18		simpl 485	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 Fn (𝐶 +o 𝐷))
19		oaordi 8503	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
20	19	ancoms 461	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
21	20	adantl 484	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
22	21	imp 409	. . . . 5 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
23		fnfvelrn 7050	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
24	18, 22, 23	syl2an2r 693	. . . 4 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
25	24	fmpttd 7085	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹)
26		simprr 780	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
27	11, 26	elmapd 8810	. . 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 259	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹 ↑m 𝐷))
29	18, 15	fnssresd 6634	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) Fn 𝐶)
30		fvex 6869	. . . . . . 7 ⊢ (𝐹‘(𝐶 +o 𝑑)) ∈ V
31		eqid 2756	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))
32	30, 31	fnmpti 6653	. . . . . 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 487	. . . . 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 43862	. . . . 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 846	. . . 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 7393	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝐶 +o 𝑑) = (𝐶 +o 𝑧))
39	38	fveq2d 6860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝐹‘(𝐶 +o 𝑑)) = (𝐹‘(𝐶 +o 𝑧)))
40		fvex 6869	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(𝐶 +o 𝑧)) ∈ V
41	39, 31, 40	fvmpt 6964	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐷 → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
42	41	ad2antlr 735	. . . . . . . . . . . . . . 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 6856	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐶 +o 𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐶 +o 𝑧)))
44	43	adantl 484	. . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 3157	. . . . . . . . . 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 776	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
51		eloni 6345	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
52	5, 51	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
53		eloni 6345	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ∈ On → Ord 𝐶)
54	53	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
55		ordeldif 43783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
56	52, 54, 55	syl2anc 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
57	56	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
58	57	biimpa 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥))
59	58	ancomd 464	. . . . . . . . . . . . . . 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 518	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 43851	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑧 ∈ 𝐷 (𝐶 +o 𝑧) = 𝑥)
63	61, 62	syl 17	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . 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 518	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 3169	. . . . . . . . . . . 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 15	. . . . . . . . . . 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 415	. . . . . . . . . 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 214	. . . . . . . . 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 4079	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) → 𝑥 ∈ (𝐶 +o 𝐷))
77		eqcom 2763	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
78		fnbrfvb 6906	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
79	77, 78	bitrid 285	. . . . . . . . . 10 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
80	18, 76, 79	syl2an 604	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
81	75, 80	bitrd 281	. . . . . . . 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 586	. . . . . . 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 5160	. . . . . 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 6020	. . . . . 6 ⊢ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)}
85	83, 84	eqtr4di 2809	. . . . 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 4116	. . . 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 2791	. . 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 5972	. . . 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 4430	. . . . . . 7 ⊢ (𝐶 ⊆ (𝐶 +o 𝐷) ↔ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
91	14, 90	sylib 220	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
92	91	adantl 484	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
93	92	reseq2d 5958	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = (𝐹 ↾ (𝐶 +o 𝐷)))
94		fnresdm 6629	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
95	94	adantr 483	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
96	93, 95	eqtrd 2791	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = 𝐹)
97	87, 89, 96	3eqtr2d 2797	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹)
98		dmres 5991	. . 3 ⊢ dom (𝐹 ↾ 𝐶) = (𝐶 ∩ dom 𝐹)
99	15, 4	sseqtrrd 3968	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ dom 𝐹)
100		dfss2 3917	. . . 4 ⊢ (𝐶 ⊆ dom 𝐹 ↔ (𝐶 ∩ dom 𝐹) = 𝐶)
101	99, 100	sylib 220	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∩ dom 𝐹) = 𝐶)
102	98, 101	eqtrid 2803	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝐹 ↾ 𝐶) = 𝐶)
103	30, 31	dmmpti 6654	. . 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 7392	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (𝑢 + 𝑣) = ((𝐹 ↾ 𝐶) + 𝑣))
106	105	eqeq1d 2758	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → ((𝑢 + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + 𝑣) = 𝐹))
107		dmeq 5872	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → dom 𝑢 = dom (𝐹 ↾ 𝐶))
108	107	eqeq1d 2758	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (dom 𝑢 = 𝐶 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
109	106, 108	3anbi12d 1452	. . 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 7393	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((𝐹 ↾ 𝐶) + 𝑣) = ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))))
111	110	eqeq1d 2758	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹))
112		dmeq 5872	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → dom 𝑣 = dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))))
113	112	eqeq1d 2758	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (dom 𝑣 = 𝐷 ↔ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷))
114	111, 113	3anbi13d 1453	. . 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 3589	. 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 1397	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))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  Vcvv 3448   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5094  {copab 5156   ↦ cmpt 5175  dom cdm 5640  ran crn 5641   ↾ cres 5642  Ord word 6334  Oncon0 6335  Fun wfun 6504   Fn wfn 6505  ⟶wf 6506  ‘cfv 6510  (class class class)co 7385   ∈ cmpo 7387   +_o coa 8422   ↑_m cmap 8796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-oadd 8429  df-map 8798

This theorem is referenced by: (None)