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Theorem dfno2 42164
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 6742 . . . . . . . . 9 (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
21adantl 482 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3 onss 7768 . . . . . . . . . 10 (𝑥 ∈ On → 𝑥 ⊆ On)
43adantr 481 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5 xpss1 5694 . . . . . . . . 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 3991 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8 velpw 4606 . . . . . . 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 6717 . . . . . . 7 (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
1110adantl 482 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12 fdm 6723 . . . . . . . 8 (𝑓:𝑥⟶{1o, 2o} → dom 𝑓 = 𝑥)
1312adantl 482 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 = 𝑥)
14 simpl 483 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ∈ On)
1513, 14eqeltrd 2833 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 ∈ On)
169, 11, 15jca32 516 . . . . 5 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
1716rexlimiva 3147 . . . 4 (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
18 simprr 771 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → dom 𝑓 ∈ On)
19 feq2 6696 . . . . . 6 (𝑥 = dom 𝑓 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝑓:dom 𝑓⟶{1o, 2o}))
2019adantl 482 . . . . 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 483 . . . . . 6 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → Fun 𝑓)
22 elpwi 4608 . . . . . 6 (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23 funssxp 6743 . . . . . . 7 ((Fun 𝑓𝑓 ⊆ (On × {1o, 2o})) ↔ (𝑓:dom 𝑓⟶{1o, 2o} ∧ dom 𝑓 ⊆ On))
2423simplbi 498 . . . . . 6 ((Fun 𝑓𝑓 ⊆ (On × {1o, 2o})) → 𝑓:dom 𝑓⟶{1o, 2o})
2521, 22, 24syl2anr 597 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
2618, 20, 25rspcedvd 3614 . . . 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 2802 . 2 {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29 df-no 27135 . 2 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30 df-rab 3433 . 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 2770 1 No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2709  wrex 3070  {crab 3432  wss 3947  𝒫 cpw 4601  {cpr 4629   × cxp 5673  dom cdm 5675  Oncon0 6361  Fun wfun 6534  wf 6536  1oc1o 8455  2oc2o 8456   No csur 27132
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-ord 6364  df-on 6365  df-fun 6542  df-fn 6543  df-f 6544  df-no 27135
This theorem is referenced by: (None)
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