Theorem smoiun 8298

Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Assertion

Ref	Expression
smoiun	⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem smoiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4932	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥))
2		smofvon 8296	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐵‘𝐴) ∈ On)
3		smoel 8297	. . . . . 6 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐵‘𝑥) ∈ (𝐵‘𝐴))
4	3	3expia 1127	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝐵‘𝑥) ∈ (𝐵‘𝐴)))
5		ontr1 6364	. . . . . 6 ⊢ ((𝐵‘𝐴) ∈ On → ((𝑦 ∈ (𝐵‘𝑥) ∧ (𝐵‘𝑥) ∈ (𝐵‘𝐴)) → 𝑦 ∈ (𝐵‘𝐴)))
6	5	expcomd 417	. . . . 5 ⊢ ((𝐵‘𝐴) ∈ On → ((𝐵‘𝑥) ∈ (𝐵‘𝐴) → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))))
7	2, 4, 6	sylsyld 61	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))))
8	7	rexlimdv 3139	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
9	1, 8	biimtrid 243	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
10	9	ssrdv 3928	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∃wrex 3064   ⊆ wss 3890  ∪ ciun 4928  dom cdm 5625  Oncon0 6317  ‘cfv 6492  Smo wsmo 8282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-tr 5187  df-id 5520  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-smo 8283

This theorem is referenced by: (None)