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Theorem tz9.12lem3 9206
Description: Lemma for tz9.12 9207. (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 6367 . . . . . . . . . 10 Fun 𝐹
3 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑅1𝑣) = (𝑅1𝑦))
43eleq2d 2878 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑦)))
54rspcev 3574 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
6 rabn0 4296 . . . . . . . . . . . . 13 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
75, 6sylibr 237 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅)
8 intex 5207 . . . . . . . . . . . 12 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
97, 8sylib 221 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
10 vex 3447 . . . . . . . . . . . 12 𝑥 ∈ V
11 eleq1w 2875 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑣)))
1211rabbidv 3430 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1312inteqd 4846 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1413eleq1d 2877 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ( {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
151dmmpt 6065 . . . . . . . . . . . . 13 dom 𝐹 = {𝑧 ∈ V ∣ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V}
1614, 15elrab2 3634 . . . . . . . . . . . 12 (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
1710, 16mpbiran 708 . . . . . . . . . . 11 (𝑥 ∈ dom 𝐹 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
189, 17sylibr 237 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ dom 𝐹)
19 funfvima 6974 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
202, 18, 19sylancr 590 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
21 tz9.12lem.1 . . . . . . . . . . 11 𝐴 ∈ V
2221, 1tz9.12lem2 9205 . . . . . . . . . 10 suc (𝐹𝐴) ∈ On
2321, 1tz9.12lem1 9204 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ On
24 onsucuni 7527 . . . . . . . . . . . 12 ((𝐹𝐴) ⊆ On → (𝐹𝐴) ⊆ suc (𝐹𝐴))
2523, 24ax-mp 5 . . . . . . . . . . 11 (𝐹𝐴) ⊆ suc (𝐹𝐴)
2625sseli 3914 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ suc (𝐹𝐴))
27 r1ord2 9198 . . . . . . . . . 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 410 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴)))
3113, 1fvmptg 6747 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V) → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3210, 31mpan 689 . . . . . . . . . . 11 ( {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
338, 32sylbi 220 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
34 ssrab2 4010 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On
35 onint 7494 . . . . . . . . . . 11 (({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3634, 35mpan 689 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3733, 36eqeltrd 2893 . . . . . . . . 9 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
38 fveq2 6649 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑅1𝑦) = (𝑅1‘(𝐹𝑥)))
3938eleq2d 2878 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → (𝑥 ∈ (𝑅1𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
404cbvrabv 3442 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1𝑦)}
4139, 40elrab2 3634 . . . . . . . . . 10 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ↔ ((𝐹𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
4241simprbi 500 . . . . . . . . 9 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
437, 37, 423syl 18 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4443adantr 484 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4530, 44sseldd 3919 . . . . . 6 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
4645exp31 423 . . . . 5 (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → (𝑥𝐴𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4746com3r 87 . . . 4 (𝑥𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4847rexlimdv 3245 . . 3 (𝑥𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴))))
4948ralimia 3129 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
50 r1suc 9187 . . . . 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 2884 . . 3 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)))
5321elpw 4504 . . 3 (𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)))
54 dfss3 3906 . . 3 (𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5552, 53, 543bitri 300 . 2 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5649, 55sylibr 237 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  c0 4246  𝒫 cpw 4500   cuni 4803   cint 4841  cmpt 5113  dom cdm 5523  cima 5526  Oncon0 6163  suc csuc 6165  Fun wfun 6322  cfv 6328  𝑅1cr1 9179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181
This theorem is referenced by:  tz9.12  9207
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