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Theorem fnse 7181
Description: Condition for the well-order in fnwe 7180 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
fnse.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
fnse.2 (𝜑𝐹:𝐴𝐵)
fnse.3 (𝜑𝑅 Se 𝐵)
fnse.4 (𝜑 → (𝐹𝑤) ∈ V)
Assertion
Ref Expression
fnse (𝜑𝑇 Se 𝐴)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑤,𝐵   𝑥,𝑤,𝑦,𝐹   𝜑,𝑤   𝑤,𝑅,𝑥,𝑦   𝑥,𝑆,𝑦   𝑤,𝑇
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑤)   𝐵(𝑥,𝑦)   𝑆(𝑤)   𝑇(𝑥,𝑦)

Proof of Theorem fnse
Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnse.2 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
21ffvelrnda 6267 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 fnse.3 . . . . . . . 8 (𝜑𝑅 Se 𝐵)
4 seex 5001 . . . . . . . 8 ((𝑅 Se 𝐵 ∧ (𝐹𝑧) ∈ 𝐵) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
53, 4sylan 487 . . . . . . 7 ((𝜑 ∧ (𝐹𝑧) ∈ 𝐵) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
62, 5syldan 486 . . . . . 6 ((𝜑𝑧𝐴) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
7 snex 4835 . . . . . 6 {(𝐹𝑧)} ∈ V
8 unexg 6857 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V ∧ {(𝐹𝑧)} ∈ V) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
96, 7, 8sylancl 693 . . . . 5 ((𝜑𝑧𝐴) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
10 imaeq2 5381 . . . . . . . . 9 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → (𝐹𝑤) = (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
1110eleq1d 2672 . . . . . . . 8 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝐹𝑤) ∈ V ↔ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1211imbi2d 329 . . . . . . 7 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝜑 → (𝐹𝑤) ∈ V) ↔ (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)))
13 fnse.4 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
1412, 13vtoclg 3239 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V → (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1514impcom 445 . . . . 5 ((𝜑 ∧ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
169, 15syldan 486 . . . 4 ((𝜑𝑧𝐴) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
17 inss2 3796 . . . . . 6 (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝑇 “ {𝑧})
18 vex 3176 . . . . . . . . 9 𝑧 ∈ V
19 vex 3176 . . . . . . . . . 10 𝑤 ∈ V
2019eliniseg 5413 . . . . . . . . 9 (𝑧 ∈ V → (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
2118, 20ax-mp 5 . . . . . . . 8 (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
22 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
23 fveq2 6103 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
2422, 23breqan12d 4599 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
2522, 23eqeqan12d 2626 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑤) = (𝐹𝑧)))
26 breq12 4588 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑆𝑦𝑤𝑆𝑧))
2725, 26anbi12d 743 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)))
2824, 27orbi12d 742 . . . . . . . . . 10 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
29 fnse.1 . . . . . . . . . 10 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
3028, 29brab2ga 5117 . . . . . . . . 9 (𝑤𝑇𝑧 ↔ ((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
311ffvelrnda 6267 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐵)
3231adantrr 749 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝐹𝑤) ∈ 𝐵)
33 breq1 4586 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝐹𝑤) → (𝑢𝑅(𝐹𝑧) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3433elrab3 3332 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3532, 34syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3635biimprd 237 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤)𝑅(𝐹𝑧) → (𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)}))
37 simpl 472 . . . . . . . . . . . . . . . 16 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) = (𝐹𝑧))
38 fvex 6113 . . . . . . . . . . . . . . . . 17 (𝐹𝑤) ∈ V
3938elsn 4140 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ {(𝐹𝑧)} ↔ (𝐹𝑤) = (𝐹𝑧))
4037, 39sylibr 223 . . . . . . . . . . . . . . 15 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)})
4140a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)}))
4236, 41orim12d 879 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)})))
43 elun 3715 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ↔ ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)}))
4442, 43syl6ibr 241 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
45 simprl 790 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝑤𝐴)
4644, 45jctild 564 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
47 ffn 5958 . . . . . . . . . . . . . 14 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
481, 47syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐴)
4948adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝐹 Fn 𝐴)
50 elpreima 6245 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ↔ (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5149, 50syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ↔ (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5246, 51sylibrd 248 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5352expimpd 627 . . . . . . . . 9 (𝜑 → (((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5430, 53syl5bi 231 . . . . . . . 8 (𝜑 → (𝑤𝑇𝑧𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5521, 54syl5bi 231 . . . . . . 7 (𝜑 → (𝑤 ∈ (𝑇 “ {𝑧}) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5655ssrdv 3574 . . . . . 6 (𝜑 → (𝑇 “ {𝑧}) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5717, 56syl5ss 3579 . . . . 5 (𝜑 → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5857adantr 480 . . . 4 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5916, 58ssexd 4733 . . 3 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
6059ralrimiva 2949 . 2 (𝜑 → ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
61 dfse2 5418 . 2 (𝑇 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
6260, 61sylibr 223 1 (𝜑𝑇 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  cun 3538  cin 3539  wss 3540  {csn 4125   class class class wbr 4583  {copab 4642   Se wse 4995  ccnv 5037  cima 5041   Fn wfn 5799  wf 5800  cfv 5804
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-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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-se 4998  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812
This theorem is referenced by:  r0weon  8718
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