Theorem tfsconcat0i 44423

Description: The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 28-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
tfsconcat0i	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))

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 tfsconcat0i

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		fnrel 6641	. . . . . . . . . . 11 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
3		reldm0 5909	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
4	2, 3	syl 18	. . . . . . . . . 10 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
5		fndm 6642	. . . . . . . . . . 11 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
6	5	eqeq1d 2762	. . . . . . . . . 10 ⊢ (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅))
7	4, 6	bitrd 282	. . . . . . . . 9 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅))
8	7	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ ↔ 𝐶 = ∅))
9		simpr 490	. . . . . . . . . 10 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → 𝐵 Fn 𝐷)
10		simpr 490	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
11	9, 10	anim12i 625	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 Fn 𝐷 ∧ 𝐷 ∈ On))
12	11	anim1i 627	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 = ∅) → ((𝐵 Fn 𝐷 ∧ 𝐷 ∈ On) ∧ 𝐶 = ∅))
13	8, 12	sylbida 604	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → ((𝐵 Fn 𝐷 ∧ 𝐷 ∈ On) ∧ 𝐶 = ∅))
14		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝐶 = ∅ → (𝐶 +o 𝐷) = (∅ +o 𝐷))
15		id 23	. . . . . . . . . . . . 13 ⊢ (𝐶 = ∅ → 𝐶 = ∅)
16	14, 15	difeq12d 4064	. . . . . . . . . . . 12 ⊢ (𝐶 = ∅ → ((𝐶 +o 𝐷) ∖ 𝐶) = ((∅ +o 𝐷) ∖ ∅))
17		dif0 4326	. . . . . . . . . . . 12 ⊢ ((∅ +o 𝐷) ∖ ∅) = (∅ +o 𝐷)
18	16, 17	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → ((𝐶 +o 𝐷) ∖ 𝐶) = (∅ +o 𝐷))
19	18	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ 𝑥 ∈ (∅ +o 𝐷)))
20		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝐶 = ∅ → (𝐶 +o 𝑧) = (∅ +o 𝑧))
21	20	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝐶 = ∅ → (𝑥 = (𝐶 +o 𝑧) ↔ 𝑥 = (∅ +o 𝑧)))
22	21	anbi1d 643	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
23	22	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ 𝐷 (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
24	19, 23	anbi12d 644	. . . . . . . . 9 ⊢ (𝐶 = ∅ → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑥 ∈ (∅ +o 𝐷) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))))
25		oa0r 8547	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ On → (∅ +o 𝐷) = 𝐷)
26	25	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝐷 ∈ On → (𝑥 ∈ (∅ +o 𝐷) ↔ 𝑥 ∈ 𝐷))
27		onelon 6387	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
28		oa0r 8547	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (∅ +o 𝑧) = 𝑧)
29	28	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑥 = (∅ +o 𝑧) ↔ 𝑥 = 𝑧))
30	29	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ On → ((𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))))
31	27, 30	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ On ∧ 𝑧 ∈ 𝐷) → ((𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))))
32	31	rexbidva 3184	. . . . . . . . . . 11 ⊢ (𝐷 ∈ On → (∃𝑧 ∈ 𝐷 (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))))
33	26, 32	anbi12d 644	. . . . . . . . . 10 ⊢ (𝐷 ∈ On → ((𝑥 ∈ (∅ +o 𝐷) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑥 ∈ 𝐷 ∧ ∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)))))
34		df-rex 3087	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧(𝑧 ∈ 𝐷 ∧ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))))
35		an12 658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑥 = 𝑧 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))))
36		eqcom 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
37	36	anbi1i 636	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑧 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑧 = 𝑥 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))))
38	35, 37	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑧 = 𝑥 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))))
39	38	exbii 1880	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧(𝑧 ∈ 𝐷 ∧ (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))) ↔ ∃𝑧(𝑧 = 𝑥 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))))
40		eleq1w 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐷 ↔ 𝑥 ∈ 𝐷))
41		fveq2 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑥 → (𝐵‘𝑧) = (𝐵‘𝑥))
42	41	eqeq2d 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑥)))
43	40, 42	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑥))))
44	43	equsexvw 2037	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑥)))
45	34, 39, 44	3bitri 300	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 = (𝐵‘𝑥)))
46	45	baib 545	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐷 → (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ 𝑦 = (𝐵‘𝑥)))
47	46	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐵 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ 𝑦 = (𝐵‘𝑥)))
48		eqcom 2767	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐵‘𝑥) ↔ (𝐵‘𝑥) = 𝑦)
49	47, 48	bitrdi 290	. . . . . . . . . . . . 13 ⊢ ((𝐵 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝐵‘𝑥) = 𝑦))
50		fnbrfvb 6935	. . . . . . . . . . . . 13 ⊢ ((𝐵 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐵‘𝑥) = 𝑦 ↔ 𝑥𝐵𝑦))
51	49, 50	bitrd 282	. . . . . . . . . . . 12 ⊢ ((𝐵 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧)) ↔ 𝑥𝐵𝑦))
52	51	pm5.32da 590	. . . . . . . . . . 11 ⊢ (𝐵 Fn 𝐷 → ((𝑥 ∈ 𝐷 ∧ ∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦)))
53		fnbr 6647	. . . . . . . . . . . . . 14 ⊢ ((𝐵 Fn 𝐷 ∧ 𝑥𝐵𝑦) → 𝑥 ∈ 𝐷)
54	53	ex 418	. . . . . . . . . . . . 13 ⊢ (𝐵 Fn 𝐷 → (𝑥𝐵𝑦 → 𝑥 ∈ 𝐷))
55	54	pm4.71rd 572	. . . . . . . . . . . 12 ⊢ (𝐵 Fn 𝐷 → (𝑥𝐵𝑦 ↔ (𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦)))
56		df-br 5103	. . . . . . . . . . . 12 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
57	55, 56	bitr3di 289	. . . . . . . . . . 11 ⊢ (𝐵 Fn 𝐷 → ((𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
58	52, 57	bitrd 282	. . . . . . . . . 10 ⊢ (𝐵 Fn 𝐷 → ((𝑥 ∈ 𝐷 ∧ ∃𝑧 ∈ 𝐷 (𝑥 = 𝑧 ∧ 𝑦 = (𝐵‘𝑧))) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
59	33, 58	sylan9bbr 520	. . . . . . . . 9 ⊢ ((𝐵 Fn 𝐷 ∧ 𝐷 ∈ On) → ((𝑥 ∈ (∅ +o 𝐷) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (∅ +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
60	24, 59	sylan9bbr 520	. . . . . . . 8 ⊢ (((𝐵 Fn 𝐷 ∧ 𝐷 ∈ On) ∧ 𝐶 = ∅) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
61	60	opabbidv 5170	. . . . . . 7 ⊢ (((𝐵 Fn 𝐷 ∧ 𝐷 ∈ On) ∧ 𝐶 = ∅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵})
62	13, 61	syl 18	. . . . . 6 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵})
63		fnrel 6641	. . . . . . . . 9 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
64		opabid2 5805	. . . . . . . . 9 ⊢ (Rel 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵} = 𝐵)
65	63, 64	syl 18	. . . . . . . 8 ⊢ (𝐵 Fn 𝐷 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵} = 𝐵)
66	65	adantl 487	. . . . . . 7 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵} = 𝐵)
67	66	ad2antrr 739	. . . . . 6 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐵} = 𝐵)
68	62, 67	eqtrd 2795	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = 𝐵)
69	1, 68	uneq12d 4168	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = (∅ ∪ 𝐵))
70		0un 4345	. . . 4 ⊢ (∅ ∪ 𝐵) = 𝐵
71	69, 70	eqtrdi 2811	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐴 = ∅) → (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐵)
72	71	ex 418	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐵))
73		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)))}))
74	73	tfsconcatun 44415	. . 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)))}))
75	74	eqeq1d 2762	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵) = 𝐵 ↔ (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = 𝐵))
76	72, 75	sylibrd 262	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  ⟨cop 4589   class class class wbr 5102  {copab 5166  dom cdm 5650  Rel wrel 5655  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   +_o coa 8474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-oadd 8481

This theorem is used by:  tfsconcat0b  44424