Theorem tfsconcatrnss12 43313

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 43306	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
3		ssun1 4201	. . . 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 4062	. . 3 ⊢ (ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵) → ran 𝐴 ⊆ ran (𝐴 + 𝐵))
6		ssun2 4202	. . . 4 ⊢ ran 𝐵 ⊆ (ran 𝐴 ∪ ran 𝐵)
7	6, 4	sseqtrrid 4062	. . 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 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  {copab 5228  dom cdm 5700  ran crn 5701  Oncon0 6397   Fn wfn 6570  ‘cfv 6575  (class class class)co 7450   ∈ cmpo 7452   +_o coa 8521

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 7772

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 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-oadd 8528

This theorem is referenced by: (None)