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Theorem infxpenc2lem3 10015

Description: Lemma for infxpenc2 10016. (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 ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
infxpenc2.3	⊢ 𝑊 = (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏))
infxpenc2.4	⊢ (𝜑 → 𝐹:(ω ↑_o 2_o)–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 ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
3		infxpenc2.3	. 2 ⊢ 𝑊 = (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏))
4		infxpenc2.4	. 2 ⊢ (𝜑 → 𝐹:(ω ↑_o 2_o)–1-1-onto→ω)
5		infxpenc2.5	. 2 ⊢ (𝜑 → (𝐹‘∅) = ∅)
6		eqid 2732	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ ((ω ↑_o 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊)))) = (𝑦 ∈ {𝑥 ∈ ((ω ↑_o 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))
7		eqid 2732	. 2 ⊢ (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑_o 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))) ∘ ^◡((ω ↑_o 2_o) CNF 𝑊)) = (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑_o 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))) ∘ ^◡((ω ↑_o 2_o) CNF 𝑊))
8		eqid 2732	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤)))))) = (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))))))
9		eqid 2732	. 2 ⊢ (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤)) = (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))
10		eqid 2732	. 2 ⊢ (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) = (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧))
11		eqid 2732	. 2 ⊢ (((ω CNF (2_o ·_o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))))))) ∘ ^◡(ω CNF (𝑊 ·_o 2_o))) = (((ω CNF (2_o ·_o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))))))) ∘ ^◡(ω CNF (𝑊 ·_o 2_o)))
12		eqid 2732	. 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 2732	. 2 ⊢ (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩) = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
14		eqid 2732	. 2 ⊢ (^◡(𝑛‘𝑏) ∘ ((((((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑_o 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))) ∘ ^◡((ω ↑_o 2_o) CNF 𝑊)) ∘ (((ω CNF (2_o ·_o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))))))) ∘ ^◡(ω CNF (𝑊 ·_o 2_o)))) ∘ (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 2_o) ↑_m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))) ∘ ^◡((ω ↑_o 2_o) CNF 𝑊)) ∘ (((ω CNF (2_o ·_o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑_m (𝑊 ·_o 2_o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡((𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((2_o ·_o 𝑤) +_o 𝑧)) ∘ ^◡(𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))))))) ∘ ^◡(ω CNF (𝑊 ·_o 2_o)))) ∘ (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 10014	1 ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 × cxp 5674 ^◡ccnv 5675 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Oncon0 6364 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ωcom 7854 1_oc1o 8458 2_oc2o 8459 +_o coa 8462 ·_o comu 8463 ↑_o coe 8464 ↑_m cmap 8819 finSupp cfsupp 9360 CNF ccnf 9655

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-seqom 8447 df-1o 8465 df-2o 8466 df-oadd 8469 df-omul 8470 df-oexp 8471 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-cnf 9656

This theorem is referenced by: infxpenc2 10016

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