Theorem tfsconcatb0 42675

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 6645	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
2		reldm0 5921	. . . . . . 7 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ dom 𝐵 = ∅))
4		fndm 6646	. . . . . . 7 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
5	4	eqeq1d 2728	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → (dom 𝐵 = ∅ ↔ 𝐷 = ∅))
6	3, 5	bitrd 279	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ 𝐷 = ∅))
7	6	ad2antlr 724	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ 𝐷 = ∅))
8		rex0 4352	. . . . . . . . . . 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 481	. . . . . . . . . . 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 327	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐷 = ∅) → ¬ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
12	11	intnand 488	. . . . . . . . 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 1923	. . . . . . . 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 5547	. . . . . . . 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 233	. . . . . . 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 4391	. . . . . . 7 ⊢ ∅ ⊆ 𝐴
17	15, 16	eqsstrdi 4031	. . . . . 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 412	. . . . 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 8467	. . . . . . . . . 10 ⊢ 1o = suc ∅
20		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 𝐷 ∈ On)
21		on0eln0 6414	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
22		df-ne 2935	. . . . . . . . . . . . 13 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
23	21, 22	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ ¬ 𝐷 = ∅))
24	23	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → ∅ ∈ 𝐷)
25		onsucss 42597	. . . . . . . . . . 11 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 → suc ∅ ⊆ 𝐷))
26	20, 24, 25	sylc 65	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → suc ∅ ⊆ 𝐷)
27	19, 26	eqsstrid 4025	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ ¬ 𝐷 = ∅) → 1o ⊆ 𝐷)
28	27	ex 412	. . . . . . . 8 ⊢ (𝐷 ∈ On → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
29	28	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
30	29	adantl 481	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (¬ 𝐷 = ∅ → 1o ⊆ 𝐷))
31		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 1o ⊆ 𝐷)
32		0lt1o 8505	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ 1o
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 1o)
34	31, 33	sseldd 3978	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∅ ∈ 𝐷)
35		oaord1 8552	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
36	35	ad2antlr 724	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (∅ ∈ 𝐷 ↔ 𝐶 ∈ (𝐶 +o 𝐷)))
37	34, 36	mpbid 231	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ (𝐶 +o 𝐷))
38		ssidd 4000	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ⊆ 𝐶)
39		oacl 8536	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
40		eloni 6368	. . . . . . . . . . . . . . 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 6368	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → Ord 𝐶)
43	42	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
44	41, 43	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
45	44	ad2antlr 724	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
46		ordeldif 42589	. . . . . . . . . . . 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 710	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
49		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ∈ On)
50	49	ad2antlr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 ∈ On)
51		oa0 8517	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → 𝐶 = (𝐶 +o ∅))
54		eqidd 2727	. . . . . . . . . . . 12 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) = (𝐵‘∅))
55	53, 54	jca 511	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅)))
56		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐶 +o 𝑧) = (𝐶 +o ∅))
57	56	eqeq2d 2737	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → (𝐶 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o ∅)))
58		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝐵‘𝑧) = (𝐵‘∅))
59	58	eqeq2d 2737	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ((𝐵‘∅) = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘∅)))
60	57, 59	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → ((𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))))
61	60	rspcev 3606	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝐷 ∧ (𝐶 = (𝐶 +o ∅) ∧ (𝐵‘∅) = (𝐵‘∅))) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
62	34, 55, 61	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → ∃𝑧 ∈ 𝐷 (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧)))
63		fvexd 6900	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 1o ⊆ 𝐷) → (𝐵‘∅) ∈ V)
64		eleq1 2815	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
65	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝐶 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
66		eqeq1 2730	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝐶 = (𝐶 +o 𝑧)))
67		eqeq1 2730	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐵‘∅) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘∅) = (𝐵‘𝑧)))
68	66, 67	bi2anan9 636	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐶 ∧ 𝑦 = (𝐵‘∅)) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝐶 = (𝐶 +o 𝑧) ∧ (𝐵‘∅) = (𝐵‘𝑧))))
69	68	rexbidv 3172	. . . . . . . . . . . . 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 630	. . . . . . . . . . . 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 5526	. . . . . . . . . . 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 583	. . . . . . . . . 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 710	. . . . . . . . 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 412	. . . . . . . 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 6376	. . . . . . . . . . . . 13 ⊢ (Ord 𝐶 → ¬ 𝐶 ∈ 𝐶)
76	42, 75	syl 17	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → ¬ 𝐶 ∈ 𝐶)
77	76	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ¬ 𝐶 ∈ 𝐶)
78	77	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ 𝐶)
79		fndm 6646	. . . . . . . . . . . 12 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
80	79	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → dom 𝐴 = 𝐶)
81	80	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐴 = 𝐶)
82	78, 81	neleqtrrd 2850	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ 𝐶 ∈ dom 𝐴)
83	49	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
84		fvexd 6900	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵‘∅) ∈ V)
85	83, 84	opeldmd 5900	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴 → 𝐶 ∈ dom 𝐴))
86	82, 85	mtod 197	. . . . . . . 8 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ¬ ⟨𝐶, (𝐵‘∅)⟩ ∈ 𝐴)
87	74, 86	jctird 526	. . . . . . 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 4042	. . . . . . 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 226	. . . 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 279	. . 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 4178	. . 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 287	. 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 42668	. . 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 2728	. 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 282	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∃wrex 3064  Vcvv 3468   ∖ cdif 3940   ∪ cun 3941   ⊆ wss 3943  ∅c0 4317  ⟨cop 4629  {copab 5203  dom cdm 5669  Rel wrel 5674  Ord word 6357  Oncon0 6358  suc csuc 6360   Fn wfn 6532  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1_oc1o 8460   +_o coa 8464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471

This theorem is referenced by: (None)