Theorem tfsconcatrev 43310

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 6759	. . . . . 6 ⊢ (𝐹 Fn (𝐶 +o 𝐷) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
2	1	biimpi 216	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
3	2	adantr 480	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
4		fndm 6682	. . . . . . . 8 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → dom 𝐹 = (𝐶 +o 𝐷))
5	4	adantr 480	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 = (𝐶 +o 𝐷))
6		oacl 8591	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
8	5, 7	eqeltrd 2844	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 ∈ On)
9		fnfun 6679	. . . . . . 7 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → Fun 𝐹)
10	9	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → Fun 𝐹)
11		funrnex 7994	. . . . . 6 ⊢ (dom 𝐹 ∈ On → (Fun 𝐹 → ran 𝐹 ∈ V))
12	8, 10, 11	sylc 65	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐹 ∈ V)
13	12, 7	elmapd 8898	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹))
14	3, 13	mpbird 257	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)))
15		oaword1 8608	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
16	15	adantl 481	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ (𝐶 +o 𝐷))
17		elmapssres 8925	. . 3 ⊢ ((𝐹 ∈ (ran 𝐹 ↑m (𝐶 +o 𝐷)) ∧ 𝐶 ⊆ (𝐶 +o 𝐷)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
18	14, 16, 17	syl2anc 583	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) ∈ (ran 𝐹 ↑m 𝐶))
19		simpl 482	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 Fn (𝐶 +o 𝐷))
20		oaordi 8602	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
21	20	ancoms 458	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
22	21	adantl 481	. . . . . 6 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
23	22	imp 406	. . . . 5 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
24		fnfvelrn 7114	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
25	19, 23, 24	syl2an2r 684	. . . 4 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
26	25	fmpttd 7149	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹)
27		simprr 772	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
28	12, 27	elmapd 8898	. . 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))
29	26, 28	mpbird 257	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹 ↑m 𝐷))
30	19, 16	fnssresd 6704	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ 𝐶) Fn 𝐶)
31		fvex 6933	. . . . . . 7 ⊢ (𝐹‘(𝐶 +o 𝑑)) ∈ V
32		eqid 2740	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))
33	31, 32	fnmpti 6723	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷
34	33	a1i 11	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷)
35		simpr 484	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
36		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)))}))
37	36	tfsconcatun 43299	. . . . 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)))}))
38	30, 34, 35, 37	syl21anc 837	. . . 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)))}))
39		oveq2 7456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝐶 +o 𝑑) = (𝐶 +o 𝑧))
40	39	fveq2d 6924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝐹‘(𝐶 +o 𝑑)) = (𝐹‘(𝐶 +o 𝑧)))
41		fvex 6933	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(𝐶 +o 𝑧)) ∈ V
42	40, 32, 41	fvmpt 7029	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐷 → ((𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
43	42	ad2antlr 726	. . . . . . . . . . . . . . 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)))
44		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐶 +o 𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐶 +o 𝑧)))
45	44	adantl 481	. . . . . . . . . . . . . . 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)))
46	43, 45	eqtr4d 2783	. . . . . . . . . . . . . 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))
47	46	eqeq2d 2751	. . . . . . . . . . . . 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)))
48	47	biimpd 229	. . . . . . . . . . . 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)))
49	48	expimpd 453	. . . . . . . . . . 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)))
50	49	rexlimdva 3161	. . . . . . . . . 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)))
51		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
52		eloni 6405	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
53	6, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
54		eloni 6405	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ∈ On → Ord 𝐶)
55	54	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
56		ordeldif 43220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
57	53, 55, 56	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
58	57	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
59	58	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥))
60	59	ancomd 461	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷)))
61	51, 60	jca 511	. . . . . . . . . . . . . 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))))
62	61	adantr 480	. . . . . . . . . . . . 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))))
63		oawordex2 43288	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑧 ∈ 𝐷 (𝐶 +o 𝑧) = 𝑥)
64	62, 63	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)
65		simpr 484	. . . . . . . . . . . . . . . 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)
66	65	eqcomd 2746	. . . . . . . . . . . . . . 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))
67	65	fveq2d 6924	. . . . . . . . . . . . . . . 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))
68	42	ad2antlr 726	. . . . . . . . . . . . . . . 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)))
69		simpllr 775	. . . . . . . . . . . . . . . 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))
70	67, 68, 69	3eqtr4rd 2791	. . . . . . . . . . . . . . 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))
71	66, 70	jca 511	. . . . . . . . . . . . . 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)))
72	71	ex 412	. . . . . . . . . . . . 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))))
73	72	reximdva 3174	. . . . . . . . . . . 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))))
74	64, 73	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)))
75	74	ex 412	. . . . . . . . . 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))))
76	50, 75	impbid 212	. . . . . . . . 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)))
77		eldifi 4154	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) → 𝑥 ∈ (𝐶 +o 𝐷))
78		eqcom 2747	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
79		fnbrfvb 6973	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
80	78, 79	bitrid 283	. . . . . . . . . 10 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
81	19, 77, 80	syl2an 595	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
82	76, 81	bitrd 279	. . . . . . . 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))
83	82	pm5.32da 578	. . . . . . 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)))
84	83	opabbidv 5232	. . . . . 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)})
85		dfres2 6070	. . . . . 6 ⊢ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)}
86	84, 85	eqtr4di 2798	. . . . 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)))
87	86	uneq2d 4191	. . . 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))))
88	38, 87	eqtrd 2780	. . 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))))
89		resundi 6023	. . . 4 ⊢ (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = ((𝐹 ↾ 𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)))
90	89	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))))
91		undif 4505	. . . . . . 7 ⊢ (𝐶 ⊆ (𝐶 +o 𝐷) ↔ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
92	15, 91	sylib 218	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
93	92	adantl 481	. . . . 5 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
94	93	reseq2d 6009	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = (𝐹 ↾ (𝐶 +o 𝐷)))
95		fnresdm 6699	. . . . 5 ⊢ (𝐹 Fn (𝐶 +o 𝐷) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
96	95	adantr 480	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
97	94, 96	eqtrd 2780	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = 𝐹)
98	88, 90, 97	3eqtr2d 2786	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹)
99		dmres 6041	. . 3 ⊢ dom (𝐹 ↾ 𝐶) = (𝐶 ∩ dom 𝐹)
100	16, 5	sseqtrrd 4050	. . . 4 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ dom 𝐹)
101		dfss2 3994	. . . 4 ⊢ (𝐶 ⊆ dom 𝐹 ↔ (𝐶 ∩ dom 𝐹) = 𝐶)
102	100, 101	sylib 218	. . 3 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∩ dom 𝐹) = 𝐶)
103	99, 102	eqtrid 2792	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝐹 ↾ 𝐶) = 𝐶)
104	31, 32	dmmpti 6724	. . 3 ⊢ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷
105	104	a1i 11	. 2 ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)
106		oveq1 7455	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (𝑢 + 𝑣) = ((𝐹 ↾ 𝐶) + 𝑣))
107	106	eqeq1d 2742	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → ((𝑢 + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + 𝑣) = 𝐹))
108		dmeq 5928	. . . . 5 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → dom 𝑢 = dom (𝐹 ↾ 𝐶))
109	108	eqeq1d 2742	. . . 4 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (dom 𝑢 = 𝐶 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
110	107, 109	3anbi12d 1437	. . 3 ⊢ (𝑢 = (𝐹 ↾ 𝐶) → (((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷) ↔ (((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ∧ dom (𝐹 ↾ 𝐶) = 𝐶 ∧ dom 𝑣 = 𝐷)))
111		oveq2 7456	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((𝐹 ↾ 𝐶) + 𝑣) = ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))))
112	111	eqeq1d 2742	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (((𝐹 ↾ 𝐶) + 𝑣) = 𝐹 ↔ ((𝐹 ↾ 𝐶) + (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹))
113		dmeq 5928	. . . . 5 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → dom 𝑣 = dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))))
114	113	eqeq1d 2742	. . . 4 ⊢ (𝑣 = (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (dom 𝑣 = 𝐷 ↔ dom (𝑑 ∈ 𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷))
115	112, 114	3anbi13d 1438	. . 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)))
116	110, 115	rspc2ev 3648	. 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))
117	18, 29, 98, 103, 105, 116	syl113anc 1382	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  {copab 5228   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ↾ cres 5702  Ord word 6394  Oncon0 6395  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   +_o coa 8519   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526  df-map 8886

This theorem is referenced by: (None)