Theorem infxpenc2lem2 10061

Description: Lemma for infxpenc2 10063. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
infxpenc2.1	⊢ (𝜑 → 𝐴 ∈ On)
infxpenc2.2	⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
infxpenc2.3	⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
infxpenc2.4	⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
infxpenc2.5	⊢ (𝜑 → (𝐹‘∅) = ∅)
infxpenc2.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
infxpenc2.h	⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
infxpenc2.l	⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
infxpenc2.x	⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
infxpenc2.y	⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
infxpenc2.j	⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
infxpenc2.z	⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
infxpenc2.t	⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
infxpenc2.g	⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))

Assertion

Ref	Expression
infxpenc2lem2	⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Distinct variable groups:   𝑔,𝑏,𝑛,𝑤,𝑥,𝑦,𝐴   𝜑,𝑏,𝑤,𝑥,𝑦   𝑧,𝑔,𝑊,𝑤,𝑥,𝑦   𝑔,𝐹,𝑥,𝑦   𝑔,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑔,𝑛)   𝐴(𝑧)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐹(𝑧,𝑤,𝑛,𝑏)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑛,𝑏)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝑊(𝑛,𝑏)   𝑋(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑌(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)

Proof of Theorem infxpenc2lem2

Step	Hyp	Ref	Expression
1		infxpenc2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
2	1	mptexd 7245	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐴 ↦ 𝐺) ∈ V)
3	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐴 ∈ On)
4		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ 𝐴)
5		onelon 6408	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On)
6	3, 4, 5	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ On)
7		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ω ⊆ 𝑏)
8		infxpenc2.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
9		infxpenc2.3	. . . . . . . 8 ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
10	1, 8, 9	infxpenc2lem1 10060	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
11	10	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑊 ∈ (On ∖ 1o))
12		infxpenc2.4	. . . . . . 7 ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
14		infxpenc2.5	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝐹‘∅) = ∅)
16	10	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊))
17		infxpenc2.k	. . . . . 6 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
18		infxpenc2.h	. . . . . 6 ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
19		infxpenc2.l	. . . . . 6 ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
20		infxpenc2.x	. . . . . 6 ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
21		infxpenc2.y	. . . . . 6 ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
22		infxpenc2.j	. . . . . 6 ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
23		infxpenc2.z	. . . . . 6 ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
24		infxpenc2.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
25		infxpenc2.g	. . . . . 6 ⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
26	6, 7, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	infxpenc 10059	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏)
27		f1of 6847	. . . . . . . . 9 ⊢ (𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏 → 𝐺:(𝑏 × 𝑏)⟶𝑏)
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)⟶𝑏)
29		vex 3483	. . . . . . . . 9 ⊢ 𝑏 ∈ V
30	29, 29	xpex 7774	. . . . . . . 8 ⊢ (𝑏 × 𝑏) ∈ V
31		fex 7247	. . . . . . . 8 ⊢ ((𝐺:(𝑏 × 𝑏)⟶𝑏 ∧ (𝑏 × 𝑏) ∈ V) → 𝐺 ∈ V)
32	28, 30, 31	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺 ∈ V)
33		eqid 2736	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐴 ↦ 𝐺) = (𝑏 ∈ 𝐴 ↦ 𝐺)
34	33	fvmpt2 7026	. . . . . . 7 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝐺 ∈ V) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏) = 𝐺)
35	4, 32, 34	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏) = 𝐺)
36	35	f1oeq1d 6842	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ 𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏))
37	26, 36	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)
38	37	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
39	38	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
40		nfmpt1 5249	. . . 4 ⊢ Ⅎ𝑏(𝑏 ∈ 𝐴 ↦ 𝐺)
41	40	nfeq2 2922	. . 3 ⊢ Ⅎ𝑏 𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺)
42		fveq1 6904	. . . . 5 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → (𝑔‘𝑏) = ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏))
43	42	f1oeq1d 6842	. . . 4 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → ((𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
44	43	imbi2d 340	. . 3 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → ((ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
45	41, 44	ralbid 3272	. 2 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → (∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
46	2, 39, 45	spcedv 3597	1 ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  {crab 3435  Vcvv 3479   ∖ cdif 3947   ⊆ wss 3950  ∅c0 4332  ⟨cop 4631   class class class wbr 5142   ↦ cmpt 5224   I cid 5576   × cxp 5682  ^◡ccnv 5683  ran crn 5685   ↾ cres 5686   ∘ ccom 5688  Oncon0 6383  ⟶wf 6556  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  ωcom 7888  1_oc1o 8500  2_oc2o 8501   +_o coa 8504   ·_o comu 8505   ↑_o coe 8506   ↑_m cmap 8867   finSupp cfsupp 9402   CNF ccnf 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-cnf 9703

This theorem is referenced by:  infxpenc2lem3  10062