Theorem tfsconcat00 43343

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 43338	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3	2	eqeq1d 2732	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) = ∅ ↔ (ran 𝐴 ∪ ran 𝐵) = ∅))
4	1	tfsconcatfn 43334	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5		fnrel 6623	. . 3 ⊢ ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) → Rel (𝐴 + 𝐵))
6		relrn0 5939	. . 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 6623	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
9		relrn0 5939	. . . . . 6 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
11		fnrel 6623	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → Rel 𝐵)
12		relrn0 5939	. . . . . 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 4411	. . . 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 3054  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915  ∅c0 4299  {copab 5172  dom cdm 5641  ran crn 5642  Rel wrel 5646  Oncon0 6335   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   +_o coa 8434

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441

This theorem is referenced by: (None)