Theorem infxpenc2lem3 9957

Description: Lemma for infxpenc2 9958. (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‘∅) = ∅)

Assertion

Ref	Expression
infxpenc2lem3	⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Distinct variable groups:   𝑔,𝑏,𝑛,𝑤,𝑥,𝐴   𝜑,𝑏,𝑤,𝑥   𝑔,𝑊,𝑤,𝑥   𝑔,𝐹,𝑥

Allowed substitution hints:   𝜑(𝑔,𝑛)   𝐹(𝑤,𝑛,𝑏)   𝑊(𝑛,𝑏)

Proof of Theorem infxpenc2lem3

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxpenc2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ On)
2		infxpenc2.2	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
3		infxpenc2.3	. 2 ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
4		infxpenc2.4	. 2 ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
5		infxpenc2.5	. 2 ⊢ (𝜑 → (𝐹‘∅) = ∅)
6		eqid 2736	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊)))) = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
7		eqid 2736	. 2 ⊢ (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))) ∘ ◡((ω ↑o 2o) CNF 𝑊)) = (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))) ∘ ◡((ω ↑o 2o) CNF 𝑊))
8		eqid 2736	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)))))) = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))
9		eqid 2736	. 2 ⊢ (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)) = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
10		eqid 2736	. 2 ⊢ (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
11		eqid 2736	. 2 ⊢ (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ ◡(ω CNF (𝑊 ·o 2o))) = (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ ◡(ω CNF (𝑊 ·o 2o)))
12		eqid 2736	. 2 ⊢ (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦)) = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
13		eqid 2736	. 2 ⊢ (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩) = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
14		eqid 2736	. 2 ⊢ (◡(𝑛‘𝑏) ∘ ((((((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))) ∘ ◡((ω ↑o 2o) CNF 𝑊)) ∘ (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ ◡(ω CNF (𝑊 ·o 2o)))) ∘ (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))) ∘ (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩))) = (◡(𝑛‘𝑏) ∘ ((((((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))) ∘ ◡((ω ↑o 2o) CNF 𝑊)) ∘ (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡((𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ ◡(𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ ◡(ω CNF (𝑊 ·o 2o)))) ∘ (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))) ∘ (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	infxpenc2lem2 9956	1 ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407   ∖ cdif 3907   ⊆ wss 3910  ∅c0 4282  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188   I cid 5530   × cxp 5631  ^◡ccnv 5632  ran crn 5634   ↾ cres 5635   ∘ ccom 5637  Oncon0 6317  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  ωcom 7802  1_oc1o 8405  2_oc2o 8406   +_o coa 8409   ·_o comu 8410   ↑_o coe 8411   ↑_m cmap 8765   finSupp cfsupp 9305   CNF ccnf 9597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-cnf 9598

This theorem is referenced by:  infxpenc2  9958