Theorem dfno2 44268

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 6730	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
2	1	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3		onss 7784	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4	3	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5		xpss1 5674	. . . . . . . . 9 ⊢ (𝑥 ⊆ On → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
6	4, 5	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑥 × {1o, 2o}) ⊆ (On × {1o, 2o}))
7	2, 6	sstrd 3941	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8		velpw 4562	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ↔ 𝑓 ⊆ (On × {1o, 2o}))
9	7, 8	sylibr 237	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ∈ 𝒫 (On × {1o, 2o}))
10		ffun 6705	. . . . . . 7 ⊢ (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
11	10	adantl 487	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12		fdm 6712	. . . . . . . 8 ⊢ (𝑓:𝑥⟶{1o, 2o} → dom 𝑓 = 𝑥)
13	12	adantl 487	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 = 𝑥)
14		simpl 488	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ∈ On)
15	13, 14	eqeltrd 2860	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 ∈ On)
16	9, 11, 15	jca32 525	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
17	16	rexlimiva 3155	. . . 4 ⊢ (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
18		simprr 785	. . . . 5 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → dom 𝑓 ∈ On)
19		feq2 6681	. . . . . 6 ⊢ (𝑥 = dom 𝑓 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝑓:dom 𝑓⟶{1o, 2o}))
20	19	adantl 487	. . . . 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 488	. . . . . 6 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → Fun 𝑓)
22		elpwi 4564	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23		funssxp 6731	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) ↔ (𝑓:dom 𝑓⟶{1o, 2o} ∧ dom 𝑓 ⊆ On))
24	23	simplbi 502	. . . . . 6 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) → 𝑓:dom 𝑓⟶{1o, 2o})
25	21, 22, 24	syl2anr 609	. . . . 5 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
26	18, 20, 25	rspcedvd 3578	. . . 4 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o})
27	17, 26	impbii 212	. . 3 ⊢ (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
28	27	abbii 2827	. 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29		df-no 27879	. 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30		df-rab 3413	. 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 2793	1 ⊢ No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  {crab 3412   ⊆ wss 3899  𝒫 cpw 4557  {cpr 4586   × cxp 5653  dom cdm 5655  Oncon0 6357  Fun wfun 6527  ⟶wf 6529  1_oc1o 8448  2_oc2o 8449   No csur 27876

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-ord 6360  df-on 6361  df-fun 6535  df-fn 6536  df-f 6537  df-no 27879

This theorem is used by: (None)