MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflem1c Structured version   Visualization version   GIF version

Theorem cantnflem1c 8531
Description: Lemma for cantnf 8537. (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 765 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝐵 ∈ On)
3 simplr 791 . 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 8510 . . . . . 6 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
84, 7mpbid 222 . . . . 5 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
98simpld 475 . . . 4 (𝜑𝐺:𝐵𝐴)
10 ffn 6004 . . . 4 (𝐺:𝐵𝐴𝐺 Fn 𝐵)
119, 10syl 17 . . 3 (𝜑𝐺 Fn 𝐵)
1211ad3antrrr 765 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝐺 Fn 𝐵)
13 oemapval.t . . . . . . 7 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
14 oemapval.f . . . . . . 7 (𝜑𝐹𝑆)
15 oemapvali.r . . . . . . 7 (𝜑𝐹𝑇𝐺)
16 oemapvali.x . . . . . . 7 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
175, 6, 1, 13, 14, 4, 15, 16oemapvali 8528 . . . . . 6 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
1817simp3d 1073 . . . . 5 (𝜑 → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
1918ad3antrrr 765 . . . 4 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)))
20 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
215, 6, 1, 13, 14, 4, 15, 16, 20cantnflem1b 8530 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
2221ad2antrr 761 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ⊆ (𝑂𝑢))
23 simprr 795 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝑂𝑢) ∈ 𝑥)
2417simp1d 1071 . . . . . . . 8 (𝜑𝑋𝐵)
25 onelon 5709 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
261, 24, 25syl2anc 692 . . . . . . 7 (𝜑𝑋 ∈ On)
2726ad3antrrr 765 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
28 onss 6940 . . . . . . . . . 10 (𝐵 ∈ On → 𝐵 ⊆ On)
291, 28syl 17 . . . . . . . . 9 (𝜑𝐵 ⊆ On)
3029sselda 3584 . . . . . . . 8 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
3130adantlr 750 . . . . . . 7 (((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) → 𝑥 ∈ On)
3231adantr 481 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
33 ontr2 5733 . . . . . 6 ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3427, 32, 33syl2anc 692 . . . . 5 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3522, 23, 34mp2and 714 . . . 4 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋𝑥)
36 eleq2 2687 . . . . . 6 (𝑤 = 𝑥 → (𝑋𝑤𝑋𝑥))
37 fveq2 6150 . . . . . . 7 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
38 fveq2 6150 . . . . . . 7 (𝑤 = 𝑥 → (𝐺𝑤) = (𝐺𝑥))
3937, 38eqeq12d 2636 . . . . . 6 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐺𝑤) ↔ (𝐹𝑥) = (𝐺𝑥)))
4036, 39imbi12d 334 . . . . 5 (𝑤 = 𝑥 → ((𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)) ↔ (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥))))
4140rspcv 3291 . . . 4 (𝑥𝐵 → (∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)) → (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥))))
423, 19, 35, 41syl3c 66 . . 3 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐹𝑥) = (𝐺𝑥))
43 simprl 793 . . 3 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐹𝑥) ≠ ∅)
4442, 43eqnetrrd 2858 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐺𝑥) ≠ ∅)
45 fvn0elsupp 7259 . 2 (((𝐵 ∈ On ∧ 𝑥𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
462, 3, 12, 44, 45syl22anc 1324 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 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3556  c0 3893   cuni 4404   class class class wbr 4615  {copab 4674   E cep 4985  ccnv 5075  dom cdm 5076  Oncon0 5684  suc csuc 5686   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607   supp csupp 7243   finSupp cfsupp 8222  OrdIsocoi 8361   CNF ccnf 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-seqom 7491  df-1o 7508  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-oi 8362  df-cnf 8506
This theorem is referenced by:  cantnflem1  8533
  Copyright terms: Public domain W3C validator