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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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