Theorem tfsconcat00 43775

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 43770	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3	2	eqeq1d 2739	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) = ∅ ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
4	1	tfsconcatfn 43766	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5		fnrel 6601	. . 3 ⊢ ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) → Rel (𝐴 + 𝐵))
6		relrn0 5929	. . 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 6601	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
9		relrn0 5929	. . . . . 6 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
11		fnrel 6601	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
12		relrn0 5929	. . . . . 6 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
14	10, 13	bi2anan9 639	. . . 4 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (ran 𝐴 = ∅ ∧ ran 𝐵 = ∅)))
15		un00 4386	. . . 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 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888  ∅c0 4274  {copab 5148  dom cdm 5631  ran crn 5632  Rel wrel 5636  Oncon0 6324   Fn wfn 6494  ‘cfv 6499  (class class class)co 7367   ∈ cmpo 7369   +_o coa 8402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409

This theorem is referenced by: (None)