Theorem tfsconcatb0 44289

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 6629	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
2		reldm0 5906	. . . . . . 7 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
3	1, 2	syl 18	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
4		fndm 6630	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
5	4	eqeq1d 2762	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (dom 𝐵 = ∅ ↔ 𝐷 = ∅))
6	3, 5	bitrd 282	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ 𝐷 = ∅))
7	6	ad2antlr 740	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ 𝐷 = ∅))
8		rex0 4307	. . . . . . . . . . 11 ⊢ ¬ ∃𝑧 ∈ ∅ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))
9		rexeq 3315	. . . . . . . . . . . 12 ⊢ (𝐷 = ∅ → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ ∅ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
10	9	adantl 487	. . . . . . . . . . 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 330	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → ¬ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
12	11	intnand 494	. . . . . . . . 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 1961	. . . . . . . 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 5525	. . . . . . . 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 237	. . . . . . 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 4349	. . . . . . 7 ⊢ ∅ ⊆ 𝐴
17	15, 16	eqsstrdi 3974	. . . . . 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 418	. . . . 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 8454	. . . . . . . . . 10 ⊢ 1o = suc ∅
20		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 𝐷 ∈ On)
21		on0eln0 6409	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
22		df-ne 2956	. . . . . . . . . . . . 13 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
23	21, 22	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ ¬ 𝐷 = ∅))
24	23	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → ∅ ∈ 𝐷)
25		onsucss 44211	. . . . . . . . . . 11 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 → suc ∅ ⊆ 𝐷))
26	20, 24, 25	sylc 66	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → suc ∅ ⊆ 𝐷)
27	19, 26	eqsstrid 3968	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 1o ⊆ 𝐷)
28	27	ex 418	. . . . . . . 8 ⊢ (𝐷 ∈ On → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
29	28	adantl 487	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
30	29	adantl 487	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
31		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 1o ⊆ 𝐷)
32		0lt1o 8490	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ 1o
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 1o)
34	31, 33	sseldd 3931	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 𝐷)
35		oaord1 8537	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
36	35	ad2antlr 740	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
37	34, 36	mpbid 235	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ (𝐶 +o 𝐷))
38		ssidd 3953	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ⊆ 𝐶)
39		oacl 8521	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
40		eloni 6361	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
41	39, 40	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
42		eloni 6361	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → Ord 𝐶)
43	42	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
44	41, 43	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
45	44	ad2antlr 740	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
46		ordeldif 44203	. . . . . . . . . . . 12 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝐶)))
47	45, 46	syl 18	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
49		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ∈ On)
50	49	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ On)
51		oa0 8502	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (𝐶 +o ∅) = 𝐶)
52	50, 51	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 +o ∅) = 𝐶)
53	52	eqcomd 2766	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 = (𝐶 +o ∅))
54		eqidd 2761	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) = (𝐵‘∅))
55	53, 54	jca 521	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅)))
56		oveq2 7416	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐶 +o 𝑧) = (𝐶 +o ∅))
57	56	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → (𝐶 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o ∅)))
58		fveq2 6873	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐵‘𝑧) = (𝐵‘∅))
59	58	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ((𝐵‘∅) = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘∅)))
60	57, 59	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → ((𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))))
61	60	rspcev 3576	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝐷 ∧ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
62	34, 55, 61	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
63		fvexd 6888	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) ∈ V)
64		eleq1 2848	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
65	64	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
66		eqeq1 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o 𝑧)))
67		eqeq1 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐵‘∅) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘𝑧)))
68	66, 67	bi2anan9 650	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧))))
69	68	rexbidv 3186	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 5503	. . . . . . . . . . 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 596	. . . . . . . . . 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 726	. . . . . . . . 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 418	. . . . . . . 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 6369	. . . . . . . . . . . . 13 ⊢ (Ord 𝐶 → ¬ 𝐶 ∈ 𝐶)
76	42, 75	syl 18	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → ¬ 𝐶 ∈ 𝐶)
77	76	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ¬ 𝐶 ∈ 𝐶)
78	77	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ 𝐶)
79		fndm 6630	. . . . . . . . . . . 12 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
80	79	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → dom 𝐴 = 𝐶)
81	80	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐴 = 𝐶)
82	78, 81	neleqtrrd 2883	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ dom 𝐴)
83	49	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
84		fvexd 6888	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵‘∅) ∈ V)
85	83, 84	opeldmd 5884	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴 → 𝐶 ∈ dom 𝐴))
86	82, 85	mtod 201	. . . . . . . 8 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ ⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴)
87	74, 86	jctird 536	. . . . . . 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 3996	. . . . . . 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 36	. . . . . 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 48	. . . . 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 230	. . . 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 282	. . 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 4134	. . 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 290	. 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 44282	. . 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 2762	. 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 285	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ∖ cdif 3895   ∪ cun 3896   ⊆ wss 3898  ∅c0 4278  ⟨cop 4589  {copab 5166  dom cdm 5647  Rel wrel 5652  Ord word 6350  Oncon0 6351  suc csuc 6353   Fn wfn 6522  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  1_oc1o 8447   +_o coa 8451

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-oadd 8458

This theorem is used by: (None)