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Theorem fnse 7339
Description: Condition for the well-order in fnwe 7338 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 6399 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 fnse.3 . . . . . . . 8 (𝜑𝑅 Se 𝐵)
4 seex 5106 . . . . . . . 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 4938 . . . . . 6 {(𝐹𝑧)} ∈ V
8 unexg 7001 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V ∧ {(𝐹𝑧)} ∈ V) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
96, 7, 8sylancl 695 . . . . 5 ((𝜑𝑧𝐴) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
10 imaeq2 5497 . . . . . . . . 9 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → (𝐹𝑤) = (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
1110eleq1d 2715 . . . . . . . 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 3297 . . . . . 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 3867 . . . . . 6 (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝑇 “ {𝑧})
18 vex 3234 . . . . . . . . 9 𝑧 ∈ V
19 vex 3234 . . . . . . . . . 10 𝑤 ∈ V
2019eliniseg 5529 . . . . . . . . 9 (𝑧 ∈ V → (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
2118, 20ax-mp 5 . . . . . . . 8 (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
22 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
23 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
2422, 23breqan12d 4701 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
2522, 23eqeqan12d 2667 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑤) = (𝐹𝑧)))
26 breq12 4690 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑆𝑦𝑤𝑆𝑧))
2725, 26anbi12d 747 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)))
2824, 27orbi12d 746 . . . . . . . . . 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, 29brab2a 5228 . . . . . . . . 9 (𝑤𝑇𝑧 ↔ ((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
311ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐵)
3231adantrr 753 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝐹𝑤) ∈ 𝐵)
33 breq1 4688 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝐹𝑤) → (𝑢𝑅(𝐹𝑧) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3433elrab3 3397 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3532, 34syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3635biimprd 238 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤)𝑅(𝐹𝑧) → (𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)}))
37 simpl 472 . . . . . . . . . . . . . . . 16 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) = (𝐹𝑧))
38 fvex 6239 . . . . . . . . . . . . . . . . 17 (𝐹𝑤) ∈ V
3938elsn 4225 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ {(𝐹𝑧)} ↔ (𝐹𝑤) = (𝐹𝑧))
4037, 39sylibr 224 . . . . . . . . . . . . . . 15 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)})
4140a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)}))
4236, 41orim12d 901 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)})))
43 elun 3786 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ↔ ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)}))
4442, 43syl6ibr 242 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
45 simprl 809 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝑤𝐴)
4644, 45jctild 565 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
47 ffn 6083 . . . . . . . . . . . . . 14 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
481, 47syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐴)
4948adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝐹 Fn 𝐴)
50 elpreima 6377 . . . . . . . . . . . 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 249 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5352expimpd 628 . . . . . . . . 9 (𝜑 → (((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5430, 53syl5bi 232 . . . . . . . 8 (𝜑 → (𝑤𝑇𝑧𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5521, 54syl5bi 232 . . . . . . 7 (𝜑 → (𝑤 ∈ (𝑇 “ {𝑧}) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5655ssrdv 3642 . . . . . 6 (𝜑 → (𝑇 “ {𝑧}) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5717, 56syl5ss 3647 . . . . 5 (𝜑 → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5857adantr 480 . . . 4 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5916, 58ssexd 4838 . . 3 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
6059ralrimiva 2995 . 2 (𝜑 → ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
61 dfse2 5534 . 2 (𝑇 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
6260, 61sylibr 224 1 (𝜑𝑇 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cun 3605  cin 3606  wss 3607  {csn 4210   class class class wbr 4685  {copab 4745   Se wse 5100  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-se 5103  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  r0weon  8873
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