Theorem tfsconcatrnss12 43381

Description: The range of the concatenation of transfinite sequences is a superset of the ranges of both sequences. Theorem 3 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-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
tfsconcatrnss12	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran 𝐴 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (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 tfsconcatrnss12

Step	Hyp	Ref	Expression
1		tfsconcat.op	. . 3 ⊢ + = (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 43374	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3		ssun1 4128	. . . 4 ⊢ ran 𝐴 ⊆ (ran 𝐴 ∪ ran 𝐵)
4		id 22	. . . 4 ⊢ (ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
5	3, 4	sseqtrrid 3978	. . 3 ⊢ (ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵) → ran 𝐴 ⊆ ran (𝐴 + 𝐵))
6		ssun2 4129	. . . 4 ⊢ ran 𝐵 ⊆ (ran 𝐴 ∪ ran 𝐵)
7	6, 4	sseqtrrid 3978	. . 3 ⊢ (ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵) → ran 𝐵 ⊆ ran (𝐴 + 𝐵))
8	5, 7	jca 511	. 2 ⊢ (ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵) → (ran 𝐴 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (𝐴 + 𝐵)))
9	2, 8	syl 17	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran 𝐴 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (𝐴 + 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {copab 5153  dom cdm 5616  ran crn 5617  Oncon0 6306   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   +_o coa 8382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389

This theorem is referenced by: (None)