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Theorem infxpenc2lem3 10039

Description: Lemma for infxpenc2 10040. (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 2725	. 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 2725	. 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 2725	. 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 2725	. 2 ⊢ (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤)) = (𝑧 ∈ 2_o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_o 𝑧) +_o 𝑤))
10		eqid 2725	. 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 2725	. 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 2725	. 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 2725	. 2 ⊢ (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩) = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
14		eqid 2725	. 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 10038	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 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ∖ cdif 3938 ⊆ wss 3941 ∅c0 4319 ⟨cop 4631 class class class wbr 5144 ↦ cmpt 5227 I cid 5570 × cxp 5671 ^◡ccnv 5672 ran crn 5674 ↾ cres 5675 ∘ ccom 5677 Oncon0 6365 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 ∈ cmpo 7415 ωcom 7865 1_oc1o 8473 2_oc2o 8474 +_o coa 8477 ·_o comu 8478 ↑_o coe 8479 ↑_m cmap 8838 finSupp cfsupp 9380 CNF ccnf 9679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-seqom 8462 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-oexp 8486 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-oi 9528 df-cnf 9680

This theorem is referenced by: infxpenc2 10040

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