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Theorem tz9.12lem1 8533
 Description: Lemma for tz9.12 8536. (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 5396 . 2 (𝐹𝐴) ⊆ ran 𝐹
2 tz9.12lem.2 . . . 4 𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
32rnmpt 5292 . . 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 3650 . . . . . . 7 {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On
6 eqvisset 3184 . . . . . . . 8 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
7 intex 4747 . . . . . . . 8 ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅ ↔ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ V)
86, 7sylibr 223 . . . . . . 7 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅)
9 oninton 6892 . . . . . . 7 (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ≠ ∅) → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ On)
105, 8, 9sylancr 694 . . . . . 6 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} ∈ On)
114, 10eqeltrd 2688 . . . . 5 (𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1211rexlimivw 3011 . . . 4 (∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)} → 𝑥 ∈ On)
1312abssi 3640 . . 3 {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}} ⊆ On
143, 13eqsstri 3598 . 2 ran 𝐹 ⊆ On
151, 14sstri 3577 1 (𝐹𝐴) ⊆ On
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   ↦ cmpt 4643  ran crn 5039   “ cima 5041  Oncon0 5640  ‘cfv 5804  𝑅1cr1 8508 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644 This theorem is referenced by:  tz9.12lem2  8534  tz9.12lem3  8535
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