Theorem tfsconcat00 43346

Description: The concatentation of two empty series results in an empty series. (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
tfsconcat00	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 + 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 tfsconcat00

Step	Hyp	Ref	Expression
1		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)))}))
2	1	tfsconcatrn 43341	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3	2	eqeq1d 2738	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) = ∅ ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
4	1	tfsconcatfn 43337	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5		fnrel 6645	. . 3 ⊢ ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) → Rel (𝐴 + 𝐵))
6		relrn0 5957	. . 3 ⊢ (Rel (𝐴 + 𝐵) → ((𝐴 + 𝐵) = ∅ ↔ ran (𝐴 + 𝐵) = ∅))
7	4, 5, 6	3syl 18	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵) = ∅ ↔ ran (𝐴 + 𝐵) = ∅))
8		fnrel 6645	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
9		relrn0 5957	. . . . . 6 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
11		fnrel 6645	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
12		relrn0 5957	. . . . . 6 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
14	10, 13	bi2anan9 638	. . . 4 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (ran 𝐴 = ∅ ∧ ran 𝐵 = ∅)))
15		un00 4425	. . . 4 ⊢ ((ran 𝐴 = ∅ ∧ ran 𝐵 = ∅) ↔ (ran 𝐴 ∪ ran 𝐵) = ∅)
16	14, 15	bitrdi 287	. . 3 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
17	16	adantr 480	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
18	3, 7, 17	3bitr4rd 312	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 + 𝐵) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929  ∅c0 4313  {copab 5186  dom cdm 5659  ran crn 5660  Rel wrel 5664  Oncon0 6357   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   +_o coa 8482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489

This theorem is referenced by: (None)