Theorem tfsconcat00 42776

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 42771	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3	2	eqeq1d 2730	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) = ∅ ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
4	1	tfsconcatfn 42767	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5		fnrel 6656	. . 3 ⊢ ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) → Rel (𝐴 + 𝐵))
6		relrn0 5972	. . 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 6656	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
9		relrn0 5972	. . . . . 6 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
11		fnrel 6656	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
12		relrn0 5972	. . . . . 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 4443	. . . 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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067  Vcvv 3471   ∖ cdif 3944   ∪ cun 3945  ∅c0 4323  {copab 5210  dom cdm 5678  ran crn 5679  Rel wrel 5683  Oncon0 6369   Fn wfn 6543  ‘cfv 6548  (class class class)co 7420   ∈ cmpo 7422   +_o coa 8483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-oadd 8490

This theorem is referenced by: (None)