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Theorem tz9.12lem3 9725
Description: Lemma for tz9.12 9726. (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 6540 . . . . . . . . . 10 Fun 𝐹
3 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑅1𝑣) = (𝑅1𝑦))
43eleq2d 2823 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑦)))
54rspcev 3581 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
6 rabn0 4345 . . . . . . . . . . . . 13 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1𝑣))
75, 6sylibr 233 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅)
8 intex 5294 . . . . . . . . . . . 12 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
97, 8sylib 217 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
10 vex 3449 . . . . . . . . . . . 12 𝑥 ∈ V
11 eleq1w 2820 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1𝑣) ↔ 𝑥 ∈ (𝑅1𝑣)))
1211rabbidv 3415 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1312inteqd 4912 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
1413eleq1d 2822 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ( {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V ↔ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
151dmmpt 6192 . . . . . . . . . . . . 13 dom 𝐹 = {𝑧 ∈ V ∣ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V}
1614, 15elrab2 3648 . . . . . . . . . . . 12 (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V))
1710, 16mpbiran 707 . . . . . . . . . . 11 (𝑥 ∈ dom 𝐹 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V)
189, 17sylibr 233 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ dom 𝐹)
19 funfvima 7180 . . . . . . . . . 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 9724 . . . . . . . . . 10 suc (𝐹𝐴) ∈ On
2321, 1tz9.12lem1 9723 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ On
24 onsucuni 7763 . . . . . . . . . . . 12 ((𝐹𝐴) ⊆ On → (𝐹𝐴) ⊆ suc (𝐹𝐴))
2523, 24ax-mp 5 . . . . . . . . . . 11 (𝐹𝐴) ⊆ suc (𝐹𝐴)
2625sseli 3940 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ suc (𝐹𝐴))
27 r1ord2 9717 . . . . . . . . . 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 407 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → (𝑅1‘(𝐹𝑥)) ⊆ (𝑅1‘suc (𝐹𝐴)))
3113, 1fvmptg 6946 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V) → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3210, 31mpan 688 . . . . . . . . . . 11 ( {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ V → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
338, 32sylbi 216 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
34 ssrab2 4037 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On
35 onint 7725 . . . . . . . . . . 11 (({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3634, 35mpan 688 . . . . . . . . . 10 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
3733, 36eqeltrd 2838 . . . . . . . . 9 ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ≠ ∅ → (𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)})
38 fveq2 6842 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑅1𝑦) = (𝑅1‘(𝐹𝑥)))
3938eleq2d 2823 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → (𝑥 ∈ (𝑅1𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
404cbvrabv 3417 . . . . . . . . . . 11 {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1𝑦)}
4139, 40elrab2 3648 . . . . . . . . . 10 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} ↔ ((𝐹𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹𝑥))))
4241simprbi 497 . . . . . . . . 9 ((𝐹𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
437, 37, 423syl 18 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4443adantr 481 . . . . . . 7 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘(𝐹𝑥)))
4530, 44sseldd 3945 . . . . . 6 (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1𝑦)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
4645exp31 420 . . . . 5 (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → (𝑥𝐴𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4746com3r 87 . . . 4 (𝑥𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))))
4847rexlimdv 3150 . . 3 (𝑥𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝑥 ∈ (𝑅1‘suc (𝐹𝐴))))
4948ralimia 3083 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
50 r1suc 9706 . . . . 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 2829 . . 3 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)))
5321elpw 4564 . . 3 (𝐴 ∈ 𝒫 (𝑅1‘suc (𝐹𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)))
54 dfss3 3932 . . 3 (𝐴 ⊆ (𝑅1‘suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5552, 53, 543bitri 296 . 2 (𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)) ↔ ∀𝑥𝐴 𝑥 ∈ (𝑅1‘suc (𝐹𝐴)))
5649, 55sylibr 233 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   cint 4907  cmpt 5188  dom cdm 5633  cima 5636  Oncon0 6317  suc csuc 6319  Fun wfun 6490  cfv 6496  𝑅1cr1 9698
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-r1 9700
This theorem is referenced by:  tz9.12  9726
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