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Theorem tz9.12lem3 9010
Description: Lemma for tz9.12 9011. (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 6224 . . . . . . . . . 10 Fun 𝐹
3 fveq2 6496 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑅1𝑣) = (𝑅1𝑦))
43eleq2d 2845 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑦)))
54rspcev 3529 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
6 rabn0 4219 . . . . . . . . . . . . 13 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
75, 6sylibr 226 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅)
8 intex 5092 . . . . . . . . . . . 12 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
97, 8sylib 210 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
10 vex 3412 . . . . . . . . . . . 12 𝑥 ∈ V
11 eleq1w 2842 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑣)))
1211rabbidv 3397 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1312inteqd 4750 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1413eleq1d 2844 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ( {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
151dmmpt 5930 . . . . . . . . . . . . 13 dom 𝐹 = {𝑧 ∈ V ∣ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V}
1614, 15elrab2 3593 . . . . . . . . . . . 12 (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
1710, 16mpbiran 696 . . . . . . . . . . 11 (𝑥 ∈ dom 𝐹 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
189, 17sylibr 226 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ dom 𝐹)
19 funfvima 6816 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
202, 18, 19sylancr 578 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
21 tz9.12lem.1 . . . . . . . . . . 11 𝐴 ∈ V
2221, 1tz9.12lem2 9009 . . . . . . . . . 10 suc (𝐹𝐴) ∈ On
2321, 1tz9.12lem1 9008 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ On
24 onsucuni 7357 . . . . . . . . . . . 12 ((𝐹𝐴) ⊆ On → (𝐹𝐴) ⊆ suc (𝐹𝐴))
2523, 24ax-mp 5 . . . . . . . . . . 11 (𝐹𝐴) ⊆ suc (𝐹𝐴)
2625sseli 3848 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ suc (𝐹𝐴))
27 r1ord2 9002 . . . . . . . . . 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 398 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴)))
3113, 1fvmptg 6591 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V) → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3210, 31mpan 677 . . . . . . . . . . 11 ( {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
338, 32sylbi 209 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
34 ssrab2 3940 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On
35 onint 7324 . . . . . . . . . . 11 (({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3634, 35mpan 677 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3733, 36eqeltrd 2860 . . . . . . . . 9 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
38 fveq2 6496 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑅1𝑦) = (𝑅1‘(𝐹𝑥)))
3938eleq2d 2845 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → (𝑥 ∈ (𝑅1𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
404cbvrabv 3406 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1𝑦)}
4139, 40elrab2 3593 . . . . . . . . . 10 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ↔ ((𝐹𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
4241simprbi 489 . . . . . . . . 9 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
437, 37, 423syl 18 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4443adantr 473 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4530, 44sseldd 3853 . . . . . 6 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
4645exp31 412 . . . . 5 (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → (𝑥𝐴𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4746com3r 87 . . . 4 (𝑥𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4847rexlimdv 3222 . . 3 (𝑥𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴))))
4948ralimia 3102 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
50 r1suc 8991 . . . . 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 2851 . . 3 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)))
5321elpw 4422 . . 3 (𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)))
54 dfss3 3841 . . 3 (𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5552, 53, 543bitri 289 . 2 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5649, 55sylibr 226 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wne 2961  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  wss 3823  c0 4172  𝒫 cpw 4416   cuni 4708   cint 4745  cmpt 5004  dom cdm 5403  cima 5406  Oncon0 6026  suc csuc 6028  Fun wfun 6179  cfv 6185  𝑅1cr1 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-r1 8985
This theorem is referenced by:  tz9.12  9011
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