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Theorem dftpos4 7907
Description: Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
dftpos4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos4
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tpos 7888 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 relcnv 5954 . . . . . . 7 Rel dom 𝐹
3 df-rel 5549 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 233 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 4141 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 5892 . . . . . 6 ((dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}) → ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
74, 5, 6mp2b 10 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
8 resss 5865 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
97, 8eqsstrri 3988 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
10 coss2 5714 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
119, 10ax-mp 5 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
121, 11eqsstri 3987 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
13 relco 6084 . . 3 Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
14 vex 3483 . . . . 5 𝑦 ∈ V
15 vex 3483 . . . . 5 𝑧 ∈ V
1614, 15opelco 5729 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧))
17 vex 3483 . . . . . . . . 9 𝑤 ∈ V
18 eleq1 2903 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19 sneq 4560 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019cnveqd 5733 . . . . . . . . . . . 12 (𝑥 = 𝑦{𝑥} = {𝑦})
2120unieqd 4838 . . . . . . . . . . 11 (𝑥 = 𝑦 {𝑥} = {𝑦})
2221eqeq2d 2835 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑧 = {𝑥} ↔ 𝑧 = {𝑦}))
2318, 22anbi12d 633 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦})))
24 eqeq1 2828 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = {𝑦} ↔ 𝑤 = {𝑦}))
2524anbi2d 631 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦})))
26 df-mpt 5133 . . . . . . . . 9 (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥})}
2714, 17, 23, 25, 26brab 5417 . . . . . . . 8 (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}))
28 simplr 768 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = {𝑦})
2917, 15breldm 5764 . . . . . . . . . . . . 13 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
3029adantl 485 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
3128, 30eqeltrrd 2917 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦} ∈ dom 𝐹)
32 elvv 5613 . . . . . . . . . . . . . 14 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33 opswap 6073 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧
3433eleq1i 2906 . . . . . . . . . . . . . . . . 17 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3515, 17opelcnv 5739 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3634, 35bitr4i 281 . . . . . . . . . . . . . . . 16 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)
37 sneq 4560 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
3837cnveqd 5733 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
3938unieqd 4838 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4039eleq1d 2900 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹 {⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
41 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹))
4240, 41bibi12d 349 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → (( {𝑦} ∈ dom 𝐹𝑦dom 𝐹) ↔ ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)))
4336, 42mpbiri 261 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4443exlimivv 1934 . . . . . . . . . . . . . 14 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4532, 44sylbi 220 . . . . . . . . . . . . 13 (𝑦 ∈ (V × V) → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4645biimpcd 252 . . . . . . . . . . . 12 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦dom 𝐹))
47 elun1 4138 . . . . . . . . . . . 12 (𝑦dom 𝐹𝑦 ∈ (dom 𝐹 ∪ {∅}))
4846, 47syl6 35 . . . . . . . . . . 11 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
4931, 48syl 17 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
50 elun2 4139 . . . . . . . . . . 11 (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
5150a1i 11 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
52 simpll 766 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
53 elun 4111 . . . . . . . . . . 11 (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5452, 53sylib 221 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5549, 51, 54mpjaod 857 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
56 simpr 488 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
5728, 56eqbrtrrd 5076 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦}𝐹𝑧)
5855, 57jca 515 . . . . . . . 8 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
5927, 58sylanb 584 . . . . . . 7 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
60 brtpos2 7894 . . . . . . . 8 (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧)))
6115, 60ax-mp 5 . . . . . . 7 (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6259, 61sylibr 237 . . . . . 6 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
63 df-br 5053 . . . . . 6 (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6462, 63sylib 221 . . . . 5 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6564exlimiv 1932 . . . 4 (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6616, 65sylbi 220 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6713, 66relssi 5647 . 2 (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ⊆ tpos 𝐹
6812, 67eqssi 3969 1 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480  cun 3917  wss 3919  c0 4276  {csn 4550  cop 4556   cuni 4824   class class class wbr 5052  cmpt 5132   × cxp 5540  ccnv 5541  dom cdm 5542  cres 5544  ccom 5546  Rel wrel 5547  tpos ctpos 7887
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-tpos 7888
This theorem is referenced by:  tposco  7919  nftpos  7923  oftpos  21061
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