Theorem ackbij2lem4 10199

Description: Lemma for ackbij2 10200. (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 6869	. . 3 ⊢ (𝑎 = 𝐵 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐵))
2	1	sseq2d 3970	. 2 ⊢ (𝑎 = 𝐵 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐵)))
3		fveq2 6869	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
4	3	sseq2d 3970	. 2 ⊢ (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏)))
5		fveq2 6869	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
6	5	sseq2d 3970	. 2 ⊢ (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
7		fveq2 6869	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
8	7	sseq2d 3970	. 2 ⊢ (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴)))
9		ssidd 3961	. 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 10198	. . . 4 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
13	12	ad2antrr 736	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
14		sstr2 3945	. . 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 7880	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584  ∪ ciun 4951   ↦ cmpt 5183   × cxp 5647  dom cdm 5649   “ cima 5652  suc csuc 6350  ‘cfv 6523  ωcom 7848  reccrdg 8382  Fincfn 8929  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-r1 9724  df-dju 9861  df-card 9899

This theorem is referenced by:  ackbij2  10200