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Theorem infxpenc2lem2 10011
Description: Lemma for infxpenc2 10013. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
Hypotheses
Ref Expression
infxpenc2.1 (πœ‘ β†’ 𝐴 ∈ On)
infxpenc2.2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
infxpenc2.3 π‘Š = (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘))
infxpenc2.4 (πœ‘ β†’ 𝐹:(Ο‰ ↑o 2o)–1-1-ontoβ†’Ο‰)
infxpenc2.5 (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
infxpenc2.k 𝐾 = (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))
infxpenc2.h 𝐻 = (((Ο‰ CNF π‘Š) ∘ 𝐾) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š))
infxpenc2.l 𝐿 = (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘(π‘Œ ∘ ◑𝑋))))
infxpenc2.x 𝑋 = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))
infxpenc2.y π‘Œ = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧))
infxpenc2.j 𝐽 = (((Ο‰ CNF (2o Β·o π‘Š)) ∘ 𝐿) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o)))
infxpenc2.z 𝑍 = (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦))
infxpenc2.t 𝑇 = (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩)
infxpenc2.g 𝐺 = (β—‘(π‘›β€˜π‘) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
Assertion
Ref Expression
infxpenc2lem2 (πœ‘ β†’ βˆƒπ‘”βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
Distinct variable groups:   𝑔,𝑏,𝑛,𝑀,π‘₯,𝑦,𝐴   πœ‘,𝑏,𝑀,π‘₯,𝑦   𝑧,𝑔,π‘Š,𝑀,π‘₯,𝑦   𝑔,𝐹,π‘₯,𝑦   𝑔,𝐺   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   πœ‘(𝑧,𝑔,𝑛)   𝐴(𝑧)   𝑇(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)   𝐹(𝑧,𝑀,𝑛,𝑏)   𝐺(π‘₯,𝑦,𝑧,𝑀,𝑛,𝑏)   𝐻(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)   𝐽(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)   𝐾(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)   𝐿(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)   π‘Š(𝑛,𝑏)   𝑋(𝑧,𝑀,𝑔,𝑛,𝑏)   π‘Œ(𝑧,𝑀,𝑔,𝑛,𝑏)   𝑍(π‘₯,𝑦,𝑧,𝑀,𝑔,𝑛,𝑏)
Proof of Theorem infxpenc2lem2
StepHypRef Expression
1 infxpenc2.1 . . 3 (πœ‘ β†’ 𝐴 ∈ On)
21mptexd 7222 . 2 (πœ‘ β†’ (𝑏 ∈ 𝐴 ↦ 𝐺) ∈ V)
31adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝐴 ∈ On)
4 simprl 769 . . . . . . 7 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝑏 ∈ 𝐴)
5 onelon 6386 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) β†’ 𝑏 ∈ On)
63, 4, 5syl2anc 584 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝑏 ∈ On)
7 simprr 771 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ Ο‰ βŠ† 𝑏)
8 infxpenc2.2 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
9 infxpenc2.3 . . . . . . . 8 π‘Š = (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘))
101, 8, 9infxpenc2lem1 10010 . . . . . . 7 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (π‘Š ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š)))
1110simpld 495 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ π‘Š ∈ (On βˆ– 1o))
12 infxpenc2.4 . . . . . . 7 (πœ‘ β†’ 𝐹:(Ο‰ ↑o 2o)–1-1-ontoβ†’Ο‰)
1312adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝐹:(Ο‰ ↑o 2o)–1-1-ontoβ†’Ο‰)
14 infxpenc2.5 . . . . . . 7 (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
1514adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (πΉβ€˜βˆ…) = βˆ…)
1610simprd 496 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š))
17 infxpenc2.k . . . . . 6 𝐾 = (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))
18 infxpenc2.h . . . . . 6 𝐻 = (((Ο‰ CNF π‘Š) ∘ 𝐾) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š))
19 infxpenc2.l . . . . . 6 𝐿 = (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘(π‘Œ ∘ ◑𝑋))))
20 infxpenc2.x . . . . . 6 𝑋 = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))
21 infxpenc2.y . . . . . 6 π‘Œ = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧))
22 infxpenc2.j . . . . . 6 𝐽 = (((Ο‰ CNF (2o Β·o π‘Š)) ∘ 𝐿) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o)))
23 infxpenc2.z . . . . . 6 𝑍 = (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦))
24 infxpenc2.t . . . . . 6 𝑇 = (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩)
25 infxpenc2.g . . . . . 6 𝐺 = (β—‘(π‘›β€˜π‘) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
266, 7, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25infxpenc 10009 . . . . 5 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝐺:(𝑏 Γ— 𝑏)–1-1-onto→𝑏)
27 f1of 6830 . . . . . . . . 9 (𝐺:(𝑏 Γ— 𝑏)–1-1-onto→𝑏 β†’ 𝐺:(𝑏 Γ— 𝑏)βŸΆπ‘)
2826, 27syl 17 . . . . . . . 8 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝐺:(𝑏 Γ— 𝑏)βŸΆπ‘)
29 vex 3478 . . . . . . . . 9 𝑏 ∈ V
3029, 29xpex 7736 . . . . . . . 8 (𝑏 Γ— 𝑏) ∈ V
31 fex 7224 . . . . . . . 8 ((𝐺:(𝑏 Γ— 𝑏)βŸΆπ‘ ∧ (𝑏 Γ— 𝑏) ∈ V) β†’ 𝐺 ∈ V)
3228, 30, 31sylancl 586 . . . . . . 7 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ 𝐺 ∈ V)
33 eqid 2732 . . . . . . . 8 (𝑏 ∈ 𝐴 ↦ 𝐺) = (𝑏 ∈ 𝐴 ↦ 𝐺)
3433fvmpt2 7006 . . . . . . 7 ((𝑏 ∈ 𝐴 ∧ 𝐺 ∈ V) β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘) = 𝐺)
354, 32, 34syl2anc 584 . . . . . 6 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘) = 𝐺)
3635f1oeq1d 6825 . . . . 5 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏 ↔ 𝐺:(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
3726, 36mpbird 256 . . . 4 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏)
3837expr 457 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐴) β†’ (Ο‰ βŠ† 𝑏 β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
3938ralrimiva 3146 . 2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
40 nfmpt1 5255 . . . 4 Ⅎ𝑏(𝑏 ∈ 𝐴 ↦ 𝐺)
4140nfeq2 2920 . . 3 Ⅎ𝑏 𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺)
42 fveq1 6887 . . . . 5 (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) β†’ (π‘”β€˜π‘) = ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘))
4342f1oeq1d 6825 . . . 4 (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) β†’ ((π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏 ↔ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
4443imbi2d 340 . . 3 (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) β†’ ((Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏) ↔ (Ο‰ βŠ† 𝑏 β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏)))
4541, 44ralbid 3270 . 2 (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) β†’ (βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏) ↔ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ ((𝑏 ∈ 𝐴 ↦ 𝐺)β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏)))
462, 39, 45spcedv 3588 1 (πœ‘ β†’ βˆƒπ‘”βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  Oncon0 6361  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Ο‰com 7851  1oc1o 8455  2oc2o 8456   +o coa 8459   Β·o comu 8460   ↑o coe 8461   ↑m cmap 8816   finSupp cfsupp 9357   CNF ccnf 9652
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seqom 8444  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-cnf 9653
This theorem is referenced by:  infxpenc2lem3  10012
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