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Theorem dfno2 43441
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 6763 . . . . . . . . 9 (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
21adantl 481 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3 onss 7805 . . . . . . . . . 10 (𝑥 ∈ On → 𝑥 ⊆ On)
43adantr 480 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5 xpss1 5704 . . . . . . . . 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 3994 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8 velpw 4605 . . . . . . 7 (𝑓 ∈ 𝒫 (On × {1o, 2o}) ↔ 𝑓 ⊆ (On × {1o, 2o}))
97, 8sylibr 234 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ∈ 𝒫 (On × {1o, 2o}))
10 ffun 6739 . . . . . . 7 (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
1110adantl 481 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12 fdm 6745 . . . . . . . 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 2841 . . . . . 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 3147 . . . 4 (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
18 simprr 773 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → dom 𝑓 ∈ On)
19 feq2 6717 . . . . . 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 4607 . . . . . 6 (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23 funssxp 6764 . . . . . . 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 597 . . . . 5 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
2618, 20, 25rspcedvd 3624 . . . 4 ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o})
2717, 26impbii 209 . . 3 (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
2827abbii 2809 . 2 {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29 df-no 27687 . 2 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30 df-rab 3437 . 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 2775 1 No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {crab 3436  wss 3951  𝒫 cpw 4600  {cpr 4628   × cxp 5683  dom cdm 5685  Oncon0 6384  Fun wfun 6555  wf 6557  1oc1o 8499  2oc2o 8500   No csur 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-fun 6563  df-fn 6564  df-f 6565  df-no 27687
This theorem is referenced by: (None)
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