Theorem tfsconcatb0 43859

Description: The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 25-Feb-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
tfsconcatb0	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧

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

Proof of Theorem tfsconcatb0

Step	Hyp	Ref	Expression
1		fnrel 6608	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
2		reldm0 5893	. . . . . . 7 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
4		fndm 6609	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
5	4	eqeq1d 2754	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (dom 𝐵 = ∅ ↔ 𝐷 = ∅))
6	3, 5	bitrd 281	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ 𝐷 = ∅))
7	6	ad2antlr 735	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ 𝐷 = ∅))
8		rex0 4303	. . . . . . . . . . 11 ⊢ ¬ ∃𝑧 ∈ ∅ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))
9		rexeq 3306	. . . . . . . . . . . 12 ⊢ (𝐷 = ∅ → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ ∅ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
10	9	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ ∅ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
11	8, 10	mtbiri 329	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → ¬ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
12	11	intnand 491	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → ¬ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
13	12	alrimivv 1938	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → ∀𝑥∀𝑦 ¬ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
14		opab0 5514	. . . . . . . 8 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
15	13, 14	sylibr 236	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = ∅)
16		0ss 4344	. . . . . . 7 ⊢ ∅ ⊆ 𝐴
17	15, 16	eqsstrdi 3971	. . . . . 6 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴)
18	17	ex 415	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 = ∅ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴))
19		df-1o 8421	. . . . . . . . . 10 ⊢ 1o = suc ∅
20		simpl 485	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 𝐷 ∈ On)
21		on0eln0 6388	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
22		df-ne 2948	. . . . . . . . . . . . 13 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
23	21, 22	bitrdi 289	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ ¬ 𝐷 = ∅))
24	23	biimpar 480	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → ∅ ∈ 𝐷)
25		onsucss 43781	. . . . . . . . . . 11 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 → suc ∅ ⊆ 𝐷))
26	20, 24, 25	sylc 65	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → suc ∅ ⊆ 𝐷)
27	19, 26	eqsstrid 3965	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 1o ⊆ 𝐷)
28	27	ex 415	. . . . . . . 8 ⊢ (𝐷 ∈ On → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
29	28	adantl 484	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
30	29	adantl 484	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
31		simpr 487	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 1o ⊆ 𝐷)
32		0lt1o 8457	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ 1o
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 1o)
34	31, 33	sseldd 3928	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 𝐷)
35		oaord1 8504	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
36	35	ad2antlr 735	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
37	34, 36	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ (𝐶 +o 𝐷))
38		ssidd 3950	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ⊆ 𝐶)
39		oacl 8488	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
40		eloni 6341	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
41	39, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
42		eloni 6341	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → Ord 𝐶)
43	42	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
44	41, 43	jca 518	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
45	44	ad2antlr 735	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
46		ordeldif 43773	. . . . . . . . . . . 12 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝐶)))
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝐶)))
48	37, 38, 47	mpbir2and 721	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
49		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ∈ On)
50	49	ad2antlr 735	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ On)
51		oa0 8469	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (𝐶 +o ∅) = 𝐶)
52	50, 51	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 +o ∅) = 𝐶)
53	52	eqcomd 2758	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 = (𝐶 +o ∅))
54		eqidd 2753	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) = (𝐵‘∅))
55	53, 54	jca 518	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅)))
56		oveq2 7389	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐶 +o 𝑧) = (𝐶 +o ∅))
57	56	eqeq2d 2763	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → (𝐶 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o ∅)))
58		fveq2 6852	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐵‘𝑧) = (𝐵‘∅))
59	58	eqeq2d 2763	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ((𝐵‘∅) = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘∅)))
60	57, 59	anbi12d 640	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → ((𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))))
61	60	rspcev 3572	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝐷 ∧ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
62	34, 55, 61	syl2anc 592	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
63		fvexd 6867	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) ∈ V)
64		eleq1 2840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
65	64	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
66		eqeq1 2756	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o 𝑧)))
67		eqeq1 2756	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐵‘∅) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘𝑧)))
68	66, 67	bi2anan9 646	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧))))
69	68	rexbidv 3176	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧))))
70	65, 69	anbi12d 640	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))))
71	70	opelopabga 5493	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ (𝐵‘∅) ∈ V) → (⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ↔ (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))))
72	50, 63, 71	syl2anc 592	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ↔ (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))))
73	48, 62, 72	mpbir2and 721	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})
74	73	ex 415	. . . . . . . 8 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (1o ⊆ 𝐷 → ⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
75		ordirr 6349	. . . . . . . . . . . . 13 ⊢ (Ord 𝐶 → ¬ 𝐶 ∈ 𝐶)
76	42, 75	syl 17	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → ¬ 𝐶 ∈ 𝐶)
77	76	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ¬ 𝐶 ∈ 𝐶)
78	77	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ 𝐶)
79		fndm 6609	. . . . . . . . . . . 12 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
80	79	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → dom 𝐴 = 𝐶)
81	80	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐴 = 𝐶)
82	78, 81	neleqtrrd 2875	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ dom 𝐴)
83	49	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
84		fvexd 6867	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵‘∅) ∈ V)
85	83, 84	opeldmd 5871	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴 → 𝐶 ∈ dom 𝐴))
86	82, 85	mtod 200	. . . . . . . 8 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ ⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴)
87	74, 86	jctird 533	. . . . . . 7 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (1o ⊆ 𝐷 → (⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ∧ ¬ ⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴)))
88		nelss 3993	. . . . . . 7 ⊢ ((⟨𝐶, (𝐵‘∅)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ∧ ¬ ⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴) → ¬ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴)
89	87, 88	syl6 35	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (1o ⊆ 𝐷 → ¬ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴))
90	30, 89	syld 47	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (¬ 𝐷 = ∅ → ¬ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴))
91	18, 90	impcon4bid 229	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴))
92	7, 91	bitrd 281	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴))
93		ssequn2 4132	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ⊆ 𝐴 ↔ (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐴)
94	92, 93	bitrdi 289	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐴))
95		tfsconcat.op	. . . 4 ⊢ + = (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)))}))
96	95	tfsconcatun 43852	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
97	96	eqeq1d 2754	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵) = 𝐴 ↔ (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐴))
98	94, 97	bitr4d 284	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1548   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  Vcvv 3444   ∖ cdif 3892   ∪ cun 3893   ⊆ wss 3895  ∅c0 4276  ⟨cop 4578  {copab 5152  dom cdm 5636  Rel wrel 5641  Ord word 6330  Oncon0 6331  suc csuc 6333   Fn wfn 6501  ‘cfv 6506  (class class class)co 7381   ∈ cmpo 7383  1_oc1o 8414   +_o coa 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-oadd 8425

This theorem is referenced by: (None)