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Theorem tz9.12lem1 9824
Description: Lemma for tz9.12 9827. (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 6090 . 2 (𝐹𝐴) ⊆ ran 𝐹
2 tz9.12lem.2 . . . 4 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
32rnmpt 5970 . . 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 4089 . . . . . . 7 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On
6 eqvisset 3497 . . . . . . . 8 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
7 intex 5349 . . . . . . . 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 7814 . . . . . . 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 2838 . . . . 5 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1211rexlimivw 3148 . . . 4 (∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1312abssi 4079 . . 3 {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}} ⊆ On
143, 13eqsstri 4029 . 2 ran 𝐹 ⊆ On
151, 14sstri 4004 1 (𝐹𝐴) ⊆ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  {cab 2711  wne 2937  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  c0 4338   cint 4950  cmpt 5230  ran crn 5689  cima 5691  Oncon0 6385  cfv 6562  𝑅1cr1 9799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389
This theorem is referenced by:  tz9.12lem2  9825  tz9.12lem3  9826
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