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Theorem infxpenc2lem1 8786
Description: Lemma for infxpenc2 8789. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
infxpenc2.1 (𝜑𝐴 ∈ On)
infxpenc2.2 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
infxpenc2.3 𝑊 = ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏))
Assertion
Ref Expression
infxpenc2lem1 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
Distinct variable groups:   𝑛,𝑏,𝑤,𝑥,𝐴   𝜑,𝑏,𝑤,𝑥   𝑤,𝑊,𝑥
Allowed substitution hints:   𝜑(𝑛)   𝑊(𝑛,𝑏)

Proof of Theorem infxpenc2lem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 infxpenc2.2 . . . 4 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
21r19.21bi 2927 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
32impr 648 . 2 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
4 simpr 477 . . 3 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
5 infxpenc2.3 . . . . . 6 𝑊 = ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏))
6 oveq2 6612 . . . . . . . . . 10 (𝑥 = 𝑤 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑤))
7 eqid 2621 . . . . . . . . . 10 (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)) = (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))
8 ovex 6632 . . . . . . . . . 10 (ω ↑𝑜 𝑤) ∈ V
96, 7, 8fvmpt 6239 . . . . . . . . 9 (𝑤 ∈ (On ∖ 1𝑜) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = (ω ↑𝑜 𝑤))
109ad2antrl 763 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = (ω ↑𝑜 𝑤))
11 f1ofo 6101 . . . . . . . . . 10 ((𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) → (𝑛𝑏):𝑏onto→(ω ↑𝑜 𝑤))
1211ad2antll 764 . . . . . . . . 9 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑛𝑏):𝑏onto→(ω ↑𝑜 𝑤))
13 forn 6075 . . . . . . . . 9 ((𝑛𝑏):𝑏onto→(ω ↑𝑜 𝑤) → ran (𝑛𝑏) = (ω ↑𝑜 𝑤))
1412, 13syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ran (𝑛𝑏) = (ω ↑𝑜 𝑤))
1510, 14eqtr4d 2658 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛𝑏))
16 ovex 6632 . . . . . . . . . . 11 (ω ↑𝑜 𝑥) ∈ V
17162a1i 12 . . . . . . . . . 10 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) → (ω ↑𝑜 𝑥) ∈ V))
18 omelon 8487 . . . . . . . . . . . . . 14 ω ∈ On
19 1onn 7664 . . . . . . . . . . . . . 14 1𝑜 ∈ ω
20 ondif2 7527 . . . . . . . . . . . . . 14 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
2118, 19, 20mpbir2an 954 . . . . . . . . . . . . 13 ω ∈ (On ∖ 2𝑜)
2221a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → ω ∈ (On ∖ 2𝑜))
23 eldifi 3710 . . . . . . . . . . . . 13 (𝑥 ∈ (On ∖ 1𝑜) → 𝑥 ∈ On)
2423ad2antrl 763 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → 𝑥 ∈ On)
25 eldifi 3710 . . . . . . . . . . . . 13 (𝑦 ∈ (On ∖ 1𝑜) → 𝑦 ∈ On)
2625ad2antll 764 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → 𝑦 ∈ On)
27 oecan 7614 . . . . . . . . . . . 12 ((ω ∈ (On ∖ 2𝑜) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦))
2822, 24, 26, 27syl3anc 1323 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦))
2928ex 450 . . . . . . . . . 10 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜)) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦)))
3017, 29dom2lem 7939 . . . . . . . . 9 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1→V)
31 f1f1orn 6105 . . . . . . . . 9 ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1→V → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)))
3230, 31syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)))
33 simprl 793 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → 𝑤 ∈ (On ∖ 1𝑜))
34 f1ocnvfv 6488 . . . . . . . 8 (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)) ∧ 𝑤 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛𝑏) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏)) = 𝑤))
3532, 33, 34syl2anc 692 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛𝑏) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏)) = 𝑤))
3615, 35mpd 15 . . . . . 6 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏)) = 𝑤)
375, 36syl5eq 2667 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → 𝑊 = 𝑤)
3837eleq1d 2683 . . . 4 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑊 ∈ (On ∖ 1𝑜) ↔ 𝑤 ∈ (On ∖ 1𝑜)))
3937oveq2d 6620 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (ω ↑𝑜 𝑊) = (ω ↑𝑜 𝑤))
40 f1oeq3 6086 . . . . 5 ((ω ↑𝑜 𝑊) = (ω ↑𝑜 𝑤) → ((𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
4139, 40syl 17 . . . 4 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
4238, 41anbi12d 746 . . 3 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)) ↔ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))))
434, 42mpbird 247 . 2 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
443, 43rexlimddv 3028 1 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  cmpt 4673  ccnv 5073  ran crn 5075  Oncon0 5682  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511
This theorem is referenced by:  infxpenc2lem2  8787
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