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Theorem tz9.12lem1 9644
Description: Lemma for tz9.12 9647. (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 6010 . 2 (𝐹𝐴) ⊆ ran 𝐹
2 tz9.12lem.2 . . . 4 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
32rnmpt 5896 . . 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 4025 . . . . . . 7 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On
6 eqvisset 3458 . . . . . . . 8 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
7 intex 5281 . . . . . . . 8 ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
86, 7sylibr 233 . . . . . . 7 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅)
9 oninton 7708 . . . . . . 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 2837 . . . . 5 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1211rexlimivw 3144 . . . 4 (∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1312abssi 4015 . . 3 {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}} ⊆ On
143, 13eqsstri 3966 . 2 ran 𝐹 ⊆ On
151, 14sstri 3941 1 (𝐹𝐴) ⊆ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {cab 2713  wne 2940  wrex 3070  {crab 3403  Vcvv 3441  wss 3898  c0 4269   cint 4894  cmpt 5175  ran crn 5621  cima 5623  Oncon0 6302  cfv 6479  𝑅1cr1 9619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6305  df-on 6306
This theorem is referenced by:  tz9.12lem2  9645  tz9.12lem3  9646
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