Theorem tfsconcat00 43309

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 43304	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3	2	eqeq1d 2742	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) = ∅ ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
4	1	tfsconcatfn 43300	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5		fnrel 6681	. . 3 ⊢ ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) → Rel (𝐴 + 𝐵))
6		relrn0 5995	. . 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 6681	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
9		relrn0 5995	. . . . . 6 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
11		fnrel 6681	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
12		relrn0 5995	. . . . . 6 ⊢ (Rel 𝐵 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 Fn 𝐷 → (𝐵 = ∅ ↔ ran 𝐵 = ∅))
14	10, 13	bi2anan9 637	. . . 4 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (ran 𝐴 = ∅ ∧ ran 𝐵 = ∅)))
15		un00 4468	. . . 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 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974  ∅c0 4352  {copab 5228  dom cdm 5700  ran crn 5701  Rel wrel 5705  Oncon0 6395   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   +_o coa 8519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526

This theorem is referenced by: (None)