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Theorem fnse 7813
 Description: Condition for the well-order in fnwe 7812 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)
32ffvelrnda 6829 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
4 seex 5483 . . . . . . 7 ((𝑅 Se 𝐵 ∧ (𝐹𝑧) ∈ 𝐵) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
51, 3, 4syl2an2r 684 . . . . . 6 ((𝜑𝑧𝐴) → {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V)
6 snex 5298 . . . . . 6 {(𝐹𝑧)} ∈ V
7 unexg 7455 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∈ V ∧ {(𝐹𝑧)} ∈ V) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
85, 6, 7sylancl 589 . . . . 5 ((𝜑𝑧𝐴) → ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V)
9 imaeq2 5893 . . . . . . . . 9 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → (𝐹𝑤) = (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
109eleq1d 2874 . . . . . . . 8 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝐹𝑤) ∈ V ↔ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1110imbi2d 344 . . . . . . 7 (𝑤 = ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) → ((𝜑 → (𝐹𝑤) ∈ V) ↔ (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)))
12 fnse.4 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
1311, 12vtoclg 3515 . . . . . 6 (({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V → (𝜑 → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V))
1413impcom 411 . . . . 5 ((𝜑 ∧ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ∈ V) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
158, 14syldan 594 . . . 4 ((𝜑𝑧𝐴) → (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})) ∈ V)
16 inss2 4156 . . . . . 6 (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝑇 “ {𝑧})
17 vex 3444 . . . . . . . . . 10 𝑤 ∈ V
1817eliniseg 5926 . . . . . . . . 9 (𝑧 ∈ V → (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
1918elv 3446 . . . . . . . 8 (𝑤 ∈ (𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
20 fveq2 6646 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
21 fveq2 6646 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
2220, 21breqan12d 5047 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
2320, 21eqeqan12d 2815 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑤) = (𝐹𝑧)))
24 breq12 5036 . . . . . . . . . . . 12 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑆𝑦𝑤𝑆𝑧))
2523, 24anbi12d 633 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)))
2622, 25orbi12d 916 . . . . . . . . . 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 5609 . . . . . . . . 9 (𝑤𝑇𝑧 ↔ ((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))))
292ffvelrnda 6829 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐵)
3029adantrr 716 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (𝐹𝑤) ∈ 𝐵)
31 breq1 5034 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝐹𝑤) → (𝑢𝑅(𝐹𝑧) ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3231elrab3 3629 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3330, 32syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ↔ (𝐹𝑤)𝑅(𝐹𝑧)))
3433biimprd 251 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → ((𝐹𝑤)𝑅(𝐹𝑧) → (𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)}))
35 simpl 486 . . . . . . . . . . . . . . . 16 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) = (𝐹𝑧))
36 fvex 6659 . . . . . . . . . . . . . . . . 17 (𝐹𝑤) ∈ V
3736elsn 4540 . . . . . . . . . . . . . . . 16 ((𝐹𝑤) ∈ {(𝐹𝑧)} ↔ (𝐹𝑤) = (𝐹𝑧))
3835, 37sylibr 237 . . . . . . . . . . . . . . 15 (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)})
3938a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧) → (𝐹𝑤) ∈ {(𝐹𝑧)}))
4034, 39orim12d 962 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)})))
41 elun 4076 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}) ↔ ((𝐹𝑤) ∈ {𝑢𝐵𝑢𝑅(𝐹𝑧)} ∨ (𝐹𝑤) ∈ {(𝐹𝑧)}))
4240, 41syl6ibr 255 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
43 simprl 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝑤𝐴)
4442, 43jctild 529 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤𝐴 ∧ (𝐹𝑤) ∈ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
452ffnd 6489 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐴)
4645adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → 𝐹 Fn 𝐴)
47 elpreima 6806 . . . . . . . . . . . 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 262 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝐴𝑧𝐴)) → (((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5049expimpd 457 . . . . . . . . 9 (𝜑 → (((𝑤𝐴𝑧𝐴) ∧ ((𝐹𝑤)𝑅(𝐹𝑧) ∨ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5128, 50syl5bi 245 . . . . . . . 8 (𝜑 → (𝑤𝑇𝑧𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5219, 51syl5bi 245 . . . . . . 7 (𝜑 → (𝑤 ∈ (𝑇 “ {𝑧}) → 𝑤 ∈ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)}))))
5352ssrdv 3921 . . . . . 6 (𝜑 → (𝑇 “ {𝑧}) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5416, 53sstrid 3926 . . . . 5 (𝜑 → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5554adantr 484 . . . 4 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ⊆ (𝐹 “ ({𝑢𝐵𝑢𝑅(𝐹𝑧)} ∪ {(𝐹𝑧)})))
5615, 55ssexd 5193 . . 3 ((𝜑𝑧𝐴) → (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
5756ralrimiva 3149 . 2 (𝜑 → ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
58 dfse2 5931 . 2 (𝑇 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑇 “ {𝑧})) ∈ V)
5957, 58sylibr 237 1 (𝜑𝑇 Se 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  {csn 4525   class class class wbr 5031  {copab 5093   Se wse 5477  ◡ccnv 5519   “ cima 5523   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-se 5480  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333 This theorem is referenced by:  r0weon  9426
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