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Theorem dfno2 43418
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 6764 . . . . . . . . 9 (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
21adantl 481 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3 onss 7804 . . . . . . . . . 10 (𝑥 ∈ On → 𝑥 ⊆ On)
43adantr 480 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5 xpss1 5708 . . . . . . . . 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 4006 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8 velpw 4610 . . . . . . 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 6740 . . . . . . 7 (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
1110adantl 481 . . . . . 6 ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12 fdm 6746 . . . . . . . 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 2839 . . . . . 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 3145 . . . 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 6718 . . . . . 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 4612 . . . . . 6 (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23 funssxp 6765 . . . . . . 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 2807 . 2 {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29 df-no 27702 . 2 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30 df-rab 3434 . 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 2773 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 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  wss 3963  𝒫 cpw 4605  {cpr 4633   × cxp 5687  dom cdm 5689  Oncon0 6386  Fun wfun 6557  wf 6559  1oc1o 8498  2oc2o 8499   No csur 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-fun 6565  df-fn 6566  df-f 6567  df-no 27702
This theorem is referenced by: (None)
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