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Theorem cantnflem1c 9684
Description: Lemma for cantnf 9690. (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 727 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝐵 ∈ On)
3 simplr 766 . 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 9663 . . . . . 6 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
84, 7mpbid 231 . . . . 5 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
98simpld 494 . . . 4 (𝜑𝐺:𝐵𝐴)
109ffnd 6712 . . 3 (𝜑𝐺 Fn 𝐵)
1110ad3antrrr 727 . 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 9683 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
1817ad2antrr 723 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ⊆ (𝑂𝑢))
19 simprr 770 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝑂𝑢) ∈ 𝑥)
205, 6, 1, 12, 13, 4, 14, 15oemapvali 9681 . . . . . . . . 9 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
2120simp1d 1139 . . . . . . . 8 (𝜑𝑋𝐵)
22 onelon 6383 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
231, 21, 22syl2anc 583 . . . . . . 7 (𝜑𝑋 ∈ On)
2423ad3antrrr 727 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
25 onss 7769 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
261, 25syl 17 . . . . . . . 8 (𝜑𝐵 ⊆ On)
2726sselda 3977 . . . . . . 7 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
2827ad4ant13 748 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
29 ontr2 6405 . . . . . 6 ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3024, 28, 29syl2anc 583 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3118, 19, 30mp2and 696 . . . 4 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋𝑥)
32 eleq2w 2811 . . . . . 6 (𝑤 = 𝑥 → (𝑋𝑤𝑋𝑥))
33 fveq2 6885 . . . . . . 7 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
34 fveq2 6885 . . . . . . 7 (𝑤 = 𝑥 → (𝐺𝑤) = (𝐺𝑥))
3533, 34eqeq12d 2742 . . . . . 6 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐺𝑤) ↔ (𝐹𝑥) = (𝐺𝑥)))
3632, 35imbi12d 344 . . . . 5 (𝑤 = 𝑥 → ((𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)) ↔ (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥))))
3720simp3d 1141 . . . . . 6 (𝜑 → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
3837ad3antrrr 727 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
3936, 38, 3rspcdva 3607 . . . 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 768 . . 3 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐹𝑥) ≠ ∅)
4240, 41eqnetrrd 3003 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐺𝑥) ≠ ∅)
43 fvn0elsupp 8165 . 2 (((𝐵 ∈ On ∧ 𝑥𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
442, 3, 11, 42, 43syl22anc 836 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 395   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  {crab 3426  wss 3943  c0 4317   cuni 4902   class class class wbr 5141  {copab 5203   E cep 5572  ccnv 5668  dom cdm 5669  Oncon0 6358  suc csuc 6360   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405   supp csupp 8146   finSupp cfsupp 9363  OrdIsocoi 9506   CNF ccnf 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-1o 8467  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-cnf 9659
This theorem is referenced by:  cantnflem1  9686
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