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Theorem tz9.12lem1 9827
Description: Lemma for tz9.12 9830. (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
tz9.12lem1 (𝐹𝐴) ⊆ On
Distinct variable group:   𝑧,𝑣,𝐴
Allowed substitution hints:   𝐹(𝑧,𝑣)

Proof of Theorem tz9.12lem1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imassrn 6089 . 2 (𝐹𝐴) ⊆ ran 𝐹
2 tz9.12lem.2 . . . 4 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
32rnmpt 5968 . . 3 ran 𝐹 = {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}}
4 id 22 . . . . . 6 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
5 ssrab2 4080 . . . . . . 7 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On
6 eqvisset 3500 . . . . . . . 8 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
7 intex 5344 . . . . . . . 8 ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
86, 7sylibr 234 . . . . . . 7 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅)
9 oninton 7815 . . . . . . 7 (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ On)
105, 8, 9sylancr 587 . . . . . 6 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ On)
114, 10eqeltrd 2841 . . . . 5 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1211rexlimivw 3151 . . . 4 (∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1312abssi 4070 . . 3 {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}} ⊆ On
143, 13eqsstri 4030 . 2 ran 𝐹 ⊆ On
151, 14sstri 3993 1 (𝐹𝐴) ⊆ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333   cint 4946  cmpt 5225  ran crn 5686  cima 5688  Oncon0 6384  cfv 6561  𝑅1cr1 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388
This theorem is referenced by:  tz9.12lem2  9828  tz9.12lem3  9829
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