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Theorem infxpenc2lem1 9430
Description: Lemma for infxpenc2 9433. (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 3173 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
32impr 458 . 2 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
4 simpr 488 . . 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 7143 . . . . . . . . . 10 (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7 eqid 2798 . . . . . . . . . 10 (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8 ovex 7168 . . . . . . . . . 10 (ω ↑o 𝑤) ∈ V
96, 7, 8fvmpt 6745 . . . . . . . . 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 6597 . . . . . . . . . 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 6568 . . . . . . . . 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 2836 . . . . . . 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 7168 . . . . . . . . . . 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 9093 . . . . . . . . . . . . 13 ω ∈ On
19 1onn 8248 . . . . . . . . . . . . 13 1o ∈ ω
20 ondif2 8110 . . . . . . . . . . . . 13 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
2118, 19, 20mpbir2an 710 . . . . . . . . . . . 12 ω ∈ (On ∖ 2o)
22 eldifi 4054 . . . . . . . . . . . . 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 4054 . . . . . . . . . . . . 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 8198 . . . . . . . . . . . 12 ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
2721, 23, 25, 26mp3an2i 1463 . . . . . . . . . . 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 416 . . . . . . . . . 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 8532 . . . . . . . . 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 6601 . . . . . . . . 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 7013 . . . . . . . 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 587 . . . . . . 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, 35syl5eq 2845 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
3736eleq1d 2874 . . . 4 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → (𝑊 ∈ (On ∖ 1o) ↔ 𝑤 ∈ (On ∖ 1o)))
3836oveq2d 7151 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤))) → (ω ↑o 𝑊) = (ω ↑o 𝑤))
3938f1oeq3d 6587 . . . 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 633 . . 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 260 . 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 3250 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 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  cdif 3878  wss 3881  cmpt 5110  ccnv 5518  ran crn 5520  Oncon0 6159  1-1wf1 6321  ontowfo 6322  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  ωcom 7560  1oc1o 8078  2oc2o 8079  o coe 8084
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091
This theorem is referenced by:  infxpenc2lem2  9431
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