MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz9.12lem3 Structured version   Visualization version   GIF version

Theorem tz9.12lem3 9827
Description: Lemma for tz9.12 9828. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
tz9.12lem.1 𝐴 ∈ V
tz9.12lem.2 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
Assertion
Ref Expression
tz9.12lem3 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐹(𝑧,𝑣)

Proof of Theorem tz9.12lem3
StepHypRef Expression
1 tz9.12lem.2 . . . . . . . . . . 11 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
21funmpt2 6607 . . . . . . . . . 10 Fun 𝐹
3 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑅1𝑣) = (𝑅1𝑦))
43eleq2d 2825 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑦)))
54rspcev 3622 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
6 rabn0 4395 . . . . . . . . . . . . 13 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
75, 6sylibr 234 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅)
8 intex 5350 . . . . . . . . . . . 12 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
97, 8sylib 218 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
10 vex 3482 . . . . . . . . . . . 12 𝑥 ∈ V
11 eleq1w 2822 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑣)))
1211rabbidv 3441 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1312inteqd 4956 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1413eleq1d 2824 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ( {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
151dmmpt 6262 . . . . . . . . . . . . 13 dom 𝐹 = {𝑧 ∈ V ∣ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V}
1614, 15elrab2 3698 . . . . . . . . . . . 12 (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
1710, 16mpbiran 709 . . . . . . . . . . 11 (𝑥 ∈ dom 𝐹 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
189, 17sylibr 234 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ dom 𝐹)
19 funfvima 7250 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
202, 18, 19sylancr 587 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
21 tz9.12lem.1 . . . . . . . . . . 11 𝐴 ∈ V
2221, 1tz9.12lem2 9826 . . . . . . . . . 10 suc (𝐹𝐴) ∈ On
2321, 1tz9.12lem1 9825 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ On
24 onsucuni 7848 . . . . . . . . . . . 12 ((𝐹𝐴) ⊆ On → (𝐹𝐴) ⊆ suc (𝐹𝐴))
2523, 24ax-mp 5 . . . . . . . . . . 11 (𝐹𝐴) ⊆ suc (𝐹𝐴)
2625sseli 3991 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ suc (𝐹𝐴))
27 r1ord2 9819 . . . . . . . . . 10 (suc (𝐹𝐴) ∈ On → ((𝐹𝑥) ∈ suc (𝐹𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴))))
2822, 26, 27mpsyl 68 . . . . . . . . 9 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴)))
2920, 28syl6 35 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → (𝑥𝐴 → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴))))
3029imp 406 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴)))
3113, 1fvmptg 7014 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V) → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3210, 31mpan 690 . . . . . . . . . . 11 ( {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
338, 32sylbi 217 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
34 ssrab2 4090 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On
35 onint 7810 . . . . . . . . . . 11 (({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3634, 35mpan 690 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3733, 36eqeltrd 2839 . . . . . . . . 9 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
38 fveq2 6907 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑅1𝑦) = (𝑅1‘(𝐹𝑥)))
3938eleq2d 2825 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → (𝑥 ∈ (𝑅1𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
404cbvrabv 3444 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1𝑦)}
4139, 40elrab2 3698 . . . . . . . . . 10 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ↔ ((𝐹𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
4241simprbi 496 . . . . . . . . 9 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
437, 37, 423syl 18 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4443adantr 480 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4530, 44sseldd 3996 . . . . . 6 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
4645exp31 419 . . . . 5 (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → (𝑥𝐴𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4746com3r 87 . . . 4 (𝑥𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4847rexlimdv 3151 . . 3 (𝑥𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴))))
4948ralimia 3078 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
50 r1suc 9808 . . . . 5 (suc (𝐹𝐴) ∈ On → (𝑅1‘suc suc (𝐹𝐴)) = 𝒫 (𝑅1‘suc (𝐹𝐴)))
5122, 50ax-mp 5 . . . 4 (𝑅1‘suc suc (𝐹𝐴)) = 𝒫 (𝑅1‘suc (𝐹𝐴))
5251eleq2i 2831 . . 3 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)))
5321elpw 4609 . . 3 (𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)))
54 dfss3 3984 . . 3 (𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5552, 53, 543bitri 297 . 2 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5649, 55sylibr 234 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951  cmpt 5231  dom cdm 5689  cima 5692  Oncon0 6386  suc csuc 6388  Fun wfun 6557  cfv 6563  𝑅1cr1 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802
This theorem is referenced by:  tz9.12  9828
  Copyright terms: Public domain W3C validator