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Theorem fnse 8119
Description: Condition for the well-order in fnwe 8118 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.3 . . . . . . 7 (𝜑𝑅 Se 𝐵)
2 fnse.2 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
32ffvelcdmda 7087 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
4 seex 5639 . . . . . . 7 ((𝑅 Se 𝐵 ∧ (𝐹𝑧) ∈ 𝐵) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
51, 3, 4syl2an2r 684 . . . . . 6 ((𝜑𝑧𝐴) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
6 snex 5432 . . . . . 6 {(𝐹𝑧)} ∈ V
7 unexg 7736 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V ∧ {(𝐹𝑧)} ∈ V) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
85, 6, 7sylancl 587 . . . . 5 ((𝜑𝑧𝐴) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
9 imaeq2 6056 . . . . . . . . 9 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → (𝐹𝑤) = (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
109eleq1d 2819 . . . . . . . 8 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝐹𝑤) ∈ V ↔ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1110imbi2d 341 . . . . . . 7 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝜑 → (𝐹𝑤) ∈ V) ↔ (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)))
12 fnse.4 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
1311, 12vtoclg 3557 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V → (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1413impcom 409 . . . . 5 ((𝜑 ∧ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
158, 14syldan 592 . . . 4 ((𝜑𝑧𝐴) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
16 inss2 4230 . . . . . 6 (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝑇 “ {𝑧})
17 vex 3479 . . . . . . . . . 10 𝑤 ∈ V
1817eliniseg 6094 . . . . . . . . 9 (𝑧 ∈ V → (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
1918elv 3481 . . . . . . . 8 (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
20 fveq2 6892 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
21 fveq2 6892 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
2220, 21breqan12d 5165 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
2320, 21eqeqan12d 2747 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑤) = (𝐹𝑧)))
24 breq12 5154 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑆𝑦𝑤𝑆𝑧))
2523, 24anbi12d 632 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)))
2622, 25orbi12d 918 . . . . . . . . . 10 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
27 fnse.1 . . . . . . . . . 10 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
2826, 27brab2a 5770 . . . . . . . . 9 (𝑤𝑇𝑧 ↔ ((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
292ffvelcdmda 7087 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐵)
3029adantrr 716 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝐹𝑤) ∈ 𝐵)
31 breq1 5152 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝐹𝑤) → (𝑢𝑅(𝐹𝑧) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3231elrab3 3685 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3330, 32syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3433biimprd 247 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤)𝑅(𝐹𝑧) → (𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)}))
35 simpl 484 . . . . . . . . . . . . . . . 16 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) = (𝐹𝑧))
36 fvex 6905 . . . . . . . . . . . . . . . . 17 (𝐹𝑤) ∈ V
3736elsn 4644 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ {(𝐹𝑧)} ↔ (𝐹𝑤) = (𝐹𝑧))
3835, 37sylibr 233 . . . . . . . . . . . . . . 15 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)})
3938a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)}))
4034, 39orim12d 964 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)})))
41 elun 4149 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ↔ ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)}))
4240, 41imbitrrdi 251 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
43 simprl 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝑤𝐴)
4442, 43jctild 527 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
452ffnd 6719 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐴)
4645adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝐹 Fn 𝐴)
47 elpreima 7060 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ↔ (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
4846, 47syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ↔ (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
4944, 48sylibrd 259 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5049expimpd 455 . . . . . . . . 9 (𝜑 → (((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5128, 50biimtrid 241 . . . . . . . 8 (𝜑 → (𝑤𝑇𝑧𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5219, 51biimtrid 241 . . . . . . 7 (𝜑 → (𝑤 ∈ (𝑇 “ {𝑧}) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5352ssrdv 3989 . . . . . 6 (𝜑 → (𝑇 “ {𝑧}) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5416, 53sstrid 3994 . . . . 5 (𝜑 → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5554adantr 482 . . . 4 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5615, 55ssexd 5325 . . 3 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
5756ralrimiva 3147 . 2 (𝜑 → ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
58 dfse2 6100 . 2 (𝑇 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
5957, 58sylibr 233 1 (𝜑𝑇 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  {crab 3433  Vcvv 3475  cun 3947  cin 3948  wss 3949  {csn 4629   class class class wbr 5149  {copab 5211   Se wse 5630  ccnv 5676  cima 5680   Fn wfn 6539  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-se 5633  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  r0weon  10007
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