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

Theorem cantnflem1c 9631
Description: Lemma for cantnf 9637. (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 9610 . . . . . 6 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
84, 7mpbid 231 . . . . 5 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
98simpld 496 . . . 4 (𝜑𝐺:𝐵𝐴)
109ffnd 6673 . . 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 9630 . . . . . 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 9628 . . . . . . . . 9 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
2120simp1d 1143 . . . . . . . 8 (𝜑𝑋𝐵)
22 onelon 6346 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
231, 21, 22syl2anc 585 . . . . . . 7 (𝜑𝑋 ∈ On)
2423ad3antrrr 729 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
25 onss 7723 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
261, 25syl 17 . . . . . . . 8 (𝜑𝐵 ⊆ On)
2726sselda 3948 . . . . . . 7 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
2827ad4ant13 750 . . . . . 6 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
29 ontr2 6368 . . . . . 6 ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂𝑢) ∧ (𝑂𝑢) ∈ 𝑥) → 𝑋𝑥))
3024, 28, 29syl2anc 585 . . . . 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 2818 . . . . . 6 (𝑤 = 𝑥 → (𝑋𝑤𝑋𝑥))
33 fveq2 6846 . . . . . . 7 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
34 fveq2 6846 . . . . . . 7 (𝑤 = 𝑥 → (𝐺𝑤) = (𝐺𝑥))
3533, 34eqeq12d 2749 . . . . . 6 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐺𝑤) ↔ (𝐹𝑥) = (𝐺𝑥)))
3632, 35imbi12d 345 . . . . 5 (𝑤 = 𝑥 → ((𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤)) ↔ (𝑋𝑥 → (𝐹𝑥) = (𝐺𝑥))))
3720simp3d 1145 . . . . . 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 3584 . . . 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 3009 . 2 ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → (𝐺𝑥) ≠ ∅)
43 fvn0elsupp 8115 . 2 (((𝐵 ∈ On ∧ 𝑥𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
442, 3, 11, 42, 43syl22anc 838 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 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  {crab 3406  wss 3914  c0 4286   cuni 4869   class class class wbr 5109  {copab 5171   E cep 5540  ccnv 5636  dom cdm 5637  Oncon0 6321  suc csuc 6323   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361   supp csupp 8096   finSupp cfsupp 9311  OrdIsocoi 9453   CNF ccnf 9605
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seqom 8398  df-1o 8416  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-oi 9454  df-cnf 9606
This theorem is referenced by:  cantnflem1  9633
  Copyright terms: Public domain W3C validator