Theorem infxpenc2lem2 10058

Description: Lemma for infxpenc2 10060. (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 7244	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐴 ↦ 𝐺) ∈ V)
3	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐴 ∈ On)
4		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ 𝐴)
5		onelon 6411	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On)
6	3, 4, 5	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ On)
7		simprr 773	. . . . . 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 10057	. . . . . . 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 10056	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏)
27		f1of 6849	. . . . . . . . 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 3482	. . . . . . . . 9 ⊢ 𝑏 ∈ V
30	29, 29	xpex 7772	. . . . . . . 8 ⊢ (𝑏 × 𝑏) ∈ V
31		fex 7246	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐴 ↦ 𝐺) = (𝑏 ∈ 𝐴 ↦ 𝐺)
34	33	fvmpt2 7027	. . . . . . 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 6844	. . . . 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 3144	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
40		nfmpt1 5256	. . . 4 ⊢ Ⅎ𝑏(𝑏 ∈ 𝐴 ↦ 𝐺)
41	40	nfeq2 2921	. . 3 ⊢ Ⅎ𝑏 𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺)
42		fveq1 6906	. . . . 5 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → (𝑔‘𝑏) = ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏))
43	42	f1oeq1d 6844	. . . 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 3271	. 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 3598	1 ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231   I cid 5582   × cxp 5687  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   ∘ ccom 5693  Oncon0 6386  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  ωcom 7887  1_oc1o 8498  2_oc2o 8499   +_o coa 8502   ·_o comu 8503   ↑_o coe 8504   ↑_m cmap 8865   finSupp cfsupp 9399   CNF ccnf 9699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700

This theorem is referenced by:  infxpenc2lem3  10059