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Theorem infxpenc2lem1 8725
 Description: Lemma for infxpenc2 8728. (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 2916 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
32impr 647 . 2 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
4 simpr 476 . . 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 6557 . . . . . . . . . 10 (𝑥 = 𝑤 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑤))
7 eqid 2610 . . . . . . . . . 10 (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)) = (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))
8 ovex 6577 . . . . . . . . . 10 (ω ↑𝑜 𝑤) ∈ V
96, 7, 8fvmpt 6191 . . . . . . . . 9 (𝑤 ∈ (On ∖ 1𝑜) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = (ω ↑𝑜 𝑤))
109ad2antrl 760 . . . . . . . 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 6057 . . . . . . . . . 10 ((𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) → (𝑛𝑏):𝑏onto→(ω ↑𝑜 𝑤))
1211ad2antll 761 . . . . . . . . 9 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑛𝑏):𝑏onto→(ω ↑𝑜 𝑤))
13 forn 6031 . . . . . . . . 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 2647 . . . . . . 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 6577 . . . . . . . . . . 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 8426 . . . . . . . . . . . . . 14 ω ∈ On
19 1onn 7606 . . . . . . . . . . . . . 14 1𝑜 ∈ ω
20 ondif2 7469 . . . . . . . . . . . . . 14 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
2118, 19, 20mpbir2an 957 . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . 13 (𝑥 ∈ (On ∖ 1𝑜) → 𝑥 ∈ On)
2423ad2antrl 760 . . . . . . . . . . . 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 3694 . . . . . . . . . . . . 13 (𝑦 ∈ (On ∖ 1𝑜) → 𝑦 ∈ On)
2625ad2antll 761 . . . . . . . . . . . 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 7556 . . . . . . . . . . . 12 ((ω ∈ (On ∖ 2𝑜) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦))
2822, 24, 26, 27syl3anc 1318 . . . . . . . . . . 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 449 . . . . . . . . . 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 7881 . . . . . . . . 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 6061 . . . . . . . . 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 790 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → 𝑤 ∈ (On ∖ 1𝑜))
34 f1ocnvfv 6434 . . . . . . . 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 691 . . . . . . 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 2656 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → 𝑊 = 𝑤)
3837eleq1d 2672 . . . 4 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (𝑊 ∈ (On ∖ 1𝑜) ↔ 𝑤 ∈ (On ∖ 1𝑜)))
3937oveq2d 6565 . . . . 5 (((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))) → (ω ↑𝑜 𝑊) = (ω ↑𝑜 𝑤))
40 f1oeq3 6042 . . . . 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 743 . . 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 246 . 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 3017 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039  Oncon0 5640  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453 This theorem is referenced by:  infxpenc2lem2  8726
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