Theorem ackbij2lem4 10312

Description: Lemma for ackbij2 10313. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypotheses

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
ackbij.g	⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))

Assertion

Ref	Expression
ackbij2lem4	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ackbij2lem4

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6922	. . 3 ⊢ (𝑎 = 𝐵 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐵))
2	1	sseq2d 4041	. 2 ⊢ (𝑎 = 𝐵 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐵)))
3		fveq2 6922	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
4	3	sseq2d 4041	. 2 ⊢ (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏)))
5		fveq2 6922	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
6	5	sseq2d 4041	. 2 ⊢ (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
7		fveq2 6922	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
8	7	sseq2d 4041	. 2 ⊢ (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴)))
9		ssidd 4032	. 2 ⊢ (𝐵 ∈ ω → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐵))
10		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11		ackbij.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))
12	10, 11	ackbij2lem3 10311	. . . 4 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
13	12	ad2antrr 725	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
14		sstr2 4015	. . 3 ⊢ ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏) → ((rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
15	13, 14	syl5com 31	. 2 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
16	2, 4, 6, 8, 9, 15	findsg 7939	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ ciun 5015   ↦ cmpt 5249   × cxp 5698  dom cdm 5700   “ cima 5703  suc csuc 6399  ‘cfv 6575  ωcom 7905  reccrdg 8467  Fincfn 9005  cardccrd 10006

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-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-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-oadd 8528  df-er 8765  df-map 8888  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-r1 9835  df-dju 9972  df-card 10010

This theorem is referenced by:  ackbij2  10313