Theorem ackbij2lem4 10343

Description: Lemma for ackbij2 10344. (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 6905	. . 3 ⊢ (𝑎 = 𝐵 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐵))
2	1	sseq2d 3945	. 2 ⊢ (𝑎 = 𝐵 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐵)))
3		fveq2 6905	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
4	3	sseq2d 3945	. 2 ⊢ (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏)))
5		fveq2 6905	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
6	5	sseq2d 3945	. 2 ⊢ (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
7		fveq2 6905	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
8	7	sseq2d 3945	. 2 ⊢ (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴)))
9		ssidd 3936	. 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 10342	. . . 4 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
13	12	ad2antrr 739	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
14		sstr2 3920	. . 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 32	. 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 7930	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ ciun 4954   ↦ cmpt 5189   × cxp 5653  dom cdm 5655   “ cima 5658  suc csuc 6379  ‘cfv 6555  ωcom 7898  reccrdg 8442  Fincfn 8998  cardccrd 10040

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-r1 9792  df-dju 10006  df-card 10044

This theorem is used by:  ackbij2  10344