Theorem alephfplem4 10018

Description: Lemma for alephfp 10019. (Contributed by NM, 5-Nov-2004.)

Hypothesis

Ref	Expression
alephfplem.1	⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)

Assertion

Ref	Expression
alephfplem4	⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

Proof of Theorem alephfplem4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8365	. . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
2		alephfplem.1	. . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
3	2	fneq1i 6587	. . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
4	1, 3	mpbir 231	. . . 4 ⊢ 𝐻 Fn ω
5	2	alephfplem3 10017	. . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ)
6	5	rgen 3054	. . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
7		ffnfv 7063	. . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ))
8	4, 6, 7	mpbir2an 712	. . 3 ⊢ 𝐻:ω⟶ran ℵ
9		ssun2 4120	. . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ)
10		fss 6676	. . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
11	8, 9, 10	mp2an 693	. 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ)
12		peano1 7831	. . 3 ⊢ ∅ ∈ ω
13	2	alephfplem1 10015	. . 3 ⊢ (𝐻‘∅) ∈ ran ℵ
14		fveq2 6832	. . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅))
15	14	eleq1d 2822	. . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
16	15	rspcev 3565	. . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)
17	12, 13, 16	mp2an 693	. 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
18		omex 9553	. . 3 ⊢ ω ∈ V
19		cardinfima 10008	. . 3 ⊢ (ω ∈ V → ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ))
20	18, 19	ax-mp 5	. 2 ⊢ ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ)
21	11, 17, 20	mp2an 693	1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851  ran crn 5623   ↾ cres 5624   “ cima 5625   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  ωcom 7808  reccrdg 8339  ℵcale 9849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-inf2 9551

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9416  df-har 9463  df-card 9852  df-aleph 9853

This theorem is referenced by:  alephfp  10019  alephfp2  10020