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Theorem infxpenc2lem1 10010
Description: Lemma for infxpenc2 10013. (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 3249 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
32impr 456 . 2 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
4 simpr 486 . . 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 7412 . . . . . . . . . 10 (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7 eqid 2733 . . . . . . . . . 10 (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8 ovex 7437 . . . . . . . . . 10 (ω ↑o 𝑤) ∈ V
96, 7, 8fvmpt 6994 . . . . . . . . 9 (𝑤 ∈ (On ∖ 1o) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
109ad2antrl 727 . . . . . . . 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 6837 . . . . . . . . . 10 ((𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤) → (𝑛𝑏):𝑏onto→(ω ↑o 𝑤))
1211ad2antll 728 . . . . . . . . 9 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → (𝑛𝑏):𝑏onto→(ω ↑o 𝑤))
13 forn 6805 . . . . . . . . 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 2776 . . . . . . 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 7437 . . . . . . . . . . 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 9637 . . . . . . . . . . . . 13 ω ∈ On
19 1onn 8635 . . . . . . . . . . . . 13 1o ∈ ω
20 ondif2 8497 . . . . . . . . . . . . 13 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
2118, 19, 20mpbir2an 710 . . . . . . . . . . . 12 ω ∈ (On ∖ 2o)
22 eldifi 4125 . . . . . . . . . . . . 13 (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
2322ad2antrl 727 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o))) → 𝑥 ∈ On)
24 eldifi 4125 . . . . . . . . . . . . 13 (𝑦 ∈ (On ∖ 1o) → 𝑦 ∈ On)
2524ad2antll 728 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o))) → 𝑦 ∈ On)
26 oecan 8585 . . . . . . . . . . . 12 ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
2721, 23, 25, 26mp3an2i 1467 . . . . . . . . . . 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))
2827ex 414 . . . . . . . . . 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)))
2917, 28dom2lem 8984 . . . . . . . . 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)
30 f1f1orn 6841 . . . . . . . . 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)))
3129, 30syl 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)))
32 simprl 770 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → 𝑤 ∈ (On ∖ 1o))
33 f1ocnvfv 7271 . . . . . . . 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))
3431, 32, 33syl2anc 585 . . . . . . 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))
3515, 34mpd 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)
365, 35eqtrid 2785 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
3736eleq1d 2819 . . . 4 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → (𝑊 ∈ (On ∖ 1o) ↔ 𝑤 ∈ (On ∖ 1o)))
3836oveq2d 7420 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → (ω ↑o 𝑊) = (ω ↑o 𝑤))
3938f1oeq3d 6827 . . . 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)))
4037, 39anbi12d 632 . . 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))))
414, 40mpbird 257 . 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)))
423, 41rexlimddv 3162 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  cdif 3944  wss 3947  cmpt 5230  ccnv 5674  ran crn 5676  Oncon0 6361  1-1wf1 6537  ontowfo 6538  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7404  ωcom 7850  1oc1o 8454  2oc2o 8455  o coe 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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-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-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-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-oexp 8467
This theorem is referenced by:  infxpenc2lem2  10011
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