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.

Step	Hyp	Ref	Expression
1		fssxp 6742	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
2	1	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3		onss 7768	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4	3	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5		xpss1 5694	. . . . . . . . 9 ⊢ (𝑥 ⊆ On → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
7	2, 6	sstrd 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}))
9	7, 8	sylibr 233	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ∈ 𝒫 (On × {1o, 2o}))
10		ffun 6717	. . . . . . 7 ⊢ (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
11	10	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12		fdm 6723	. . . . . . . 8 ⊢ (𝑓:𝑥⟶{1o, 2o} → dom 𝑓 = 𝑥)
13	12	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 = 𝑥)
14		simpl 483	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ∈ On)
15	13, 14	eqeltrd 2833	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 ∈ On)
16	9, 11, 15	jca32 516	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
17	16	rexlimiva 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}))
20	19	adantl 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 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) ↔ (𝑓:dom 𝑓⟶{1o, 2o} ∧ dom 𝑓 ⊆ On))
24	23	simplbi 498	. . . . . 6 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) → 𝑓:dom 𝑓⟶{1o, 2o})
25	21, 22, 24	syl2anr 597	. . . . 5 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
26	18, 20, 25	rspcedvd 3614	. . . 4 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o})
27	17, 26	impbii 208	. . 3 ⊢ (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
28	27	abbii 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))}
31	28, 29, 30	3eqtr4i 2770	1 ⊢ No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}

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  1_oc1o 8455  2_oc2o 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)