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Theorem cantnflem1c 9138
 Description: Lemma for cantnf 9144. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
oemapval.f (𝜑𝐹𝑆)
oemapval.g (𝜑𝐺𝑆)
oemapvali.r (𝜑𝐹𝑇𝐺)
oemapvali.x 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
cantnflem1.o 𝑂 = OrdIso( E , (𝐺 supp ∅))
Assertion
Ref Expression
cantnflem1c ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
Distinct variable groups:   𝑢,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑢   𝑢,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑢,𝑥,𝑦,𝑧   𝐺,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑢,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑢,𝑥,𝑦,𝑧   𝑢,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1c
StepHypRef Expression
1 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
21ad3antrrr 729 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝐵 ∈ On)
3 simplr 768 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥𝐵)
4 oemapval.g . . . . . 6 (𝜑𝐺𝑆)
5 cantnfs.s . . . . . . 7 𝑆 = dom (𝐴 CNF 𝐵)
6 cantnfs.a . . . . . . 7 (𝜑𝐴 ∈ On)
75, 6, 1cantnfs 9117 . . . . . 6 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
84, 7mpbid 235 . . . . 5 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
98simpld 498 . . . 4 (𝜑𝐺:𝐵𝐴)
109ffnd 6495 . . 3 (𝜑𝐺 Fn 𝐵)
1110ad3antrrr 729 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝐺 Fn 𝐵)
12 oemapval.t . . . . . . 7 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
13 oemapval.f . . . . . . 7 (𝜑𝐹𝑆)
14 oemapvali.r . . . . . . 7 (𝜑𝐹𝑇𝐺)
15 oemapvali.x . . . . . . 7 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
16 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
175, 6, 1, 12, 13, 4, 14, 15, 16cantnflem1b 9137 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
1817ad2antrr 725 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ⊆ (𝑂𝑢))
19 simprr 772 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝑂𝑢) ∈ 𝑥)
205, 6, 1, 12, 13, 4, 14, 15oemapvali 9135 . . . . . . . . 9 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
2120simp1d 1139 . . . . . . . 8 (𝜑𝑋𝐵)
22 onelon 6194 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
231, 21, 22syl2anc 587 . . . . . . 7 (𝜑𝑋 ∈ On)
2423ad3antrrr 729 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
25 onss 7490 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
261, 25syl 17 . . . . . . . 8 (𝜑𝐵 ⊆ On)
2726sselda 3942 . . . . . . 7 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
2827ad4ant13 750 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
29 ontr2 6216 . . . . . 6 ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3024, 28, 29syl2anc 587 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3118, 19, 30mp2and 698 . . . 4 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋𝑥)
32 eleq2w 2897 . . . . . 6 (𝑤 = 𝑥 → (𝑋𝑤𝑋𝑥))
33 fveq2 6652 . . . . . . 7 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
34 fveq2 6652 . . . . . . 7 (𝑤 = 𝑥 → (𝐺𝑤) = (𝐺𝑥))
3533, 34eqeq12d 2838 . . . . . 6 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐺𝑤) ↔ (𝐹𝑥) = (𝐺𝑥)))
3632, 35imbi12d 348 . . . . 5 (𝑤 = 𝑥 → ((𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)) ↔ (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥))))
3720simp3d 1141 . . . . . 6 (𝜑 → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
3837ad3antrrr 729 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
3936, 38, 3rspcdva 3600 . . . 4 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥)))
4031, 39mpd 15 . . 3 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐹𝑥) = (𝐺𝑥))
41 simprl 770 . . 3 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐹𝑥) ≠ ∅)
4240, 41eqnetrrd 3079 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐺𝑥) ≠ ∅)
43 fvn0elsupp 7833 . 2 (((𝐵 ∈ On ∧ 𝑥𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
442, 3, 11, 42, 43syl22anc 837 1 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  ∃wrex 3131  {crab 3134   ⊆ wss 3908  ∅c0 4265  ∪ cuni 4813   class class class wbr 5042  {copab 5104   E cep 5441  ◡ccnv 5531  dom cdm 5532  Oncon0 6169  suc csuc 6171   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   supp csupp 7817   finSupp cfsupp 8821  OrdIsocoi 8961   CNF ccnf 9112 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113 This theorem is referenced by:  cantnflem1  9140
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