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Theorem infxpenc2lem1 9962
Description: Lemma for infxpenc2 9965. (Contributed by Mario Carneiro, 30-May-2015.)
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‘))
Assertion
Ref Expression
infxpenc2lem1 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (π‘Š ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š)))
Distinct variable groups:   𝑛,𝑏,𝑀,π‘₯,𝐴   πœ‘,𝑏,𝑀,π‘₯   𝑀,π‘Š,π‘₯
Allowed substitution hints:   πœ‘(𝑛)   π‘Š(𝑛,𝑏)

Proof of Theorem infxpenc2lem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 infxpenc2.2 . . . 4 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
21r19.21bi 3237 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐴) β†’ (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
32impr 456 . 2 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))
4 simpr 486 . . 3 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
5 infxpenc2.3 . . . . . 6 π‘Š = (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘))
6 oveq2 7370 . . . . . . . . . 10 (π‘₯ = 𝑀 β†’ (Ο‰ ↑o π‘₯) = (Ο‰ ↑o 𝑀))
7 eqid 2737 . . . . . . . . . 10 (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)) = (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))
8 ovex 7395 . . . . . . . . . 10 (Ο‰ ↑o 𝑀) ∈ V
96, 7, 8fvmpt 6953 . . . . . . . . 9 (𝑀 ∈ (On βˆ– 1o) β†’ ((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜π‘€) = (Ο‰ ↑o 𝑀))
109ad2antrl 727 . . . . . . . 8 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜π‘€) = (Ο‰ ↑o 𝑀))
11 f1ofo 6796 . . . . . . . . . 10 ((π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀) β†’ (π‘›β€˜π‘):𝑏–ontoβ†’(Ο‰ ↑o 𝑀))
1211ad2antll 728 . . . . . . . . 9 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘›β€˜π‘):𝑏–ontoβ†’(Ο‰ ↑o 𝑀))
13 forn 6764 . . . . . . . . 9 ((π‘›β€˜π‘):𝑏–ontoβ†’(Ο‰ ↑o 𝑀) β†’ ran (π‘›β€˜π‘) = (Ο‰ ↑o 𝑀))
1412, 13syl 17 . . . . . . . 8 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ran (π‘›β€˜π‘) = (Ο‰ ↑o 𝑀))
1510, 14eqtr4d 2780 . . . . . . 7 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜π‘€) = ran (π‘›β€˜π‘))
16 ovex 7395 . . . . . . . . . . 11 (Ο‰ ↑o π‘₯) ∈ V
17162a1i 12 . . . . . . . . . 10 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘₯ ∈ (On βˆ– 1o) β†’ (Ο‰ ↑o π‘₯) ∈ V))
18 omelon 9589 . . . . . . . . . . . . 13 Ο‰ ∈ On
19 1onn 8591 . . . . . . . . . . . . 13 1o ∈ Ο‰
20 ondif2 8453 . . . . . . . . . . . . 13 (Ο‰ ∈ (On βˆ– 2o) ↔ (Ο‰ ∈ On ∧ 1o ∈ Ο‰))
2118, 19, 20mpbir2an 710 . . . . . . . . . . . 12 Ο‰ ∈ (On βˆ– 2o)
22 eldifi 4091 . . . . . . . . . . . . 13 (π‘₯ ∈ (On βˆ– 1o) β†’ π‘₯ ∈ On)
2322ad2antrl 727 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) ∧ (π‘₯ ∈ (On βˆ– 1o) ∧ 𝑦 ∈ (On βˆ– 1o))) β†’ π‘₯ ∈ On)
24 eldifi 4091 . . . . . . . . . . . . 13 (𝑦 ∈ (On βˆ– 1o) β†’ 𝑦 ∈ On)
2524ad2antll 728 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) ∧ (π‘₯ ∈ (On βˆ– 1o) ∧ 𝑦 ∈ (On βˆ– 1o))) β†’ 𝑦 ∈ On)
26 oecan 8541 . . . . . . . . . . . 12 ((Ο‰ ∈ (On βˆ– 2o) ∧ π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ ((Ο‰ ↑o π‘₯) = (Ο‰ ↑o 𝑦) ↔ π‘₯ = 𝑦))
2721, 23, 25, 26mp3an2i 1467 . . . . . . . . . . 11 ((((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) ∧ (π‘₯ ∈ (On βˆ– 1o) ∧ 𝑦 ∈ (On βˆ– 1o))) β†’ ((Ο‰ ↑o π‘₯) = (Ο‰ ↑o 𝑦) ↔ π‘₯ = 𝑦))
2827ex 414 . . . . . . . . . 10 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ((π‘₯ ∈ (On βˆ– 1o) ∧ 𝑦 ∈ (On βˆ– 1o)) β†’ ((Ο‰ ↑o π‘₯) = (Ο‰ ↑o 𝑦) ↔ π‘₯ = 𝑦)))
2917, 28dom2lem 8939 . . . . . . . . 9 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)):(On βˆ– 1o)–1-1β†’V)
30 f1f1orn 6800 . . . . . . . . 9 ((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)):(On βˆ– 1o)–1-1β†’V β†’ (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)):(On βˆ– 1o)–1-1-ontoβ†’ran (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)))
3129, 30syl 17 . . . . . . . 8 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)):(On βˆ– 1o)–1-1-ontoβ†’ran (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)))
32 simprl 770 . . . . . . . 8 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ 𝑀 ∈ (On βˆ– 1o))
33 f1ocnvfv 7229 . . . . . . . 8 (((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)):(On βˆ– 1o)–1-1-ontoβ†’ran (π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯)) ∧ 𝑀 ∈ (On βˆ– 1o)) β†’ (((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜π‘€) = ran (π‘›β€˜π‘) β†’ (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘)) = 𝑀))
3431, 32, 33syl2anc 585 . . . . . . 7 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (((π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜π‘€) = ran (π‘›β€˜π‘) β†’ (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘)) = 𝑀))
3515, 34mpd 15 . . . . . 6 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘)) = 𝑀)
365, 35eqtrid 2789 . . . . 5 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ π‘Š = 𝑀)
3736eleq1d 2823 . . . 4 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘Š ∈ (On βˆ– 1o) ↔ 𝑀 ∈ (On βˆ– 1o)))
3836oveq2d 7378 . . . . 5 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (Ο‰ ↑o π‘Š) = (Ο‰ ↑o 𝑀))
3938f1oeq3d 6786 . . . 4 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ((π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š) ↔ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
4037, 39anbi12d 632 . . 3 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ ((π‘Š ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š)) ↔ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))))
414, 40mpbird 257 . 2 (((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) ∧ (𝑀 ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀))) β†’ (π‘Š ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š)))
423, 41rexlimddv 3159 1 ((πœ‘ ∧ (𝑏 ∈ 𝐴 ∧ Ο‰ βŠ† 𝑏)) β†’ (π‘Š ∈ (On βˆ– 1o) ∧ (π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915   ↦ cmpt 5193  β—‘ccnv 5637  ran crn 5639  Oncon0 6322  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  Ο‰com 7807  1oc1o 8410  2oc2o 8411   ↑o coe 8416
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677  ax-inf2 9584
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-omul 8422  df-oexp 8423
This theorem is referenced by:  infxpenc2lem2  9963
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