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Theorem dfno2 42729
Description: A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
Assertion
Ref Expression
dfno2 No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}

Proof of Theorem dfno2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fssxp 6736 . . . . . . . . 9 (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
21adantl 481 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3 onss 7766 . . . . . . . . . 10 (𝑥 ∈ On → 𝑥 ⊆ On)
43adantr 480 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5 xpss1 5686 . . . . . . . . 9 (𝑥 ⊆ On → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
64, 5syl 17 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
72, 6sstrd 3985 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8 velpw 4600 . . . . . . 7 (𝑓 ∈ 𝒫 (On × {1o, 2o}) ↔ 𝑓 ⊆ (On × {1o, 2o}))
97, 8sylibr 233 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ∈ 𝒫 (On × {1o, 2o}))
10 ffun 6711 . . . . . . 7 (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
1110adantl 481 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12 fdm 6717 . . . . . . . 8 (𝑓:𝑥⟶{1o, 2o} → dom 𝑓 = 𝑥)
1312adantl 481 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 = 𝑥)
14 simpl 482 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ∈ On)
1513, 14eqeltrd 2825 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 ∈ On)
169, 11, 15jca32 515 . . . . 5 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
1716rexlimiva 3139 . . . 4 (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
18 simprr 770 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → dom 𝑓 ∈ On)
19 feq2 6690 . . . . . 6 (𝑥 = dom 𝑓 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝑓:dom 𝑓⟶{1o, 2o}))
2019adantl 481 . . . . 5 (((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) ∧ 𝑥 = dom 𝑓) → (𝑓:𝑥⟶{1o, 2o} ↔ 𝑓:dom 𝑓⟶{1o, 2o}))
21 simpl 482 . . . . . 6 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → Fun 𝑓)
22 elpwi 4602 . . . . . 6 (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23 funssxp 6737 . . . . . . 7 ((Fun 𝑓𝑓 ⊆ (On × {1o, 2o})) ↔ (𝑓:dom 𝑓⟶{1o, 2o} ∧ dom 𝑓 ⊆ On))
2423simplbi 497 . . . . . 6 ((Fun 𝑓𝑓 ⊆ (On × {1o, 2o})) → 𝑓:dom 𝑓⟶{1o, 2o})
2521, 22, 24syl2anr 596 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
2618, 20, 25rspcedvd 3606 . . . 4 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o})
2717, 26impbii 208 . . 3 (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
2827abbii 2794 . 2 {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29 df-no 27517 . 2 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30 df-rab 3425 . 2 {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
3128, 29, 303eqtr4i 2762 1 No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2701  wrex 3062  {crab 3424  wss 3941  𝒫 cpw 4595  {cpr 4623   × cxp 5665  dom cdm 5667  Oncon0 6355  Fun wfun 6528  wf 6530  1oc1o 8455  2oc2o 8456   No csur 27514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-ord 6358  df-on 6359  df-fun 6536  df-fn 6537  df-f 6538  df-no 27517
This theorem is referenced by: (None)
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