Theorem ackbij2lem4 10132

Description: Lemma for ackbij2 10133. (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 6822	. . 3 ⊢ (𝑎 = 𝐵 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐵))
2	1	sseq2d 3962	. 2 ⊢ (𝑎 = 𝐵 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐵)))
3		fveq2 6822	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
4	3	sseq2d 3962	. 2 ⊢ (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑏)))
5		fveq2 6822	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
6	5	sseq2d 3962	. 2 ⊢ (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘suc 𝑏)))
7		fveq2 6822	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
8	7	sseq2d 3962	. 2 ⊢ (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝑎) ↔ (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴)))
9		ssidd 3953	. 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 10131	. . . 4 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
13	12	ad2antrr 726	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘suc 𝑏))
14		sstr2 3936	. . 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 7827	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573  ∪ ciun 4939   ↦ cmpt 5170   × cxp 5612  dom cdm 5614   “ cima 5617  suc csuc 6308  ‘cfv 6481  ωcom 7796  reccrdg 8328  Fincfn 8869  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-r1 9657  df-dju 9794  df-card 9832

This theorem is referenced by:  ackbij2  10133