Theorem dfno2 43942

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 6704	. . . . . . . . 9 ⊢ (𝑓:𝑥⟶{1o, 2o} → 𝑓 ⊆ (𝑥 × {1o, 2o}))
2	1	adantl 484	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (𝑥 × {1o, 2o}))
3		onss 7753	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4	3	adantr 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ⊆ On)
5		xpss1 5655	. . . . . . . . 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 3937	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
8		velpw 4550	. . . . . . 7 ⊢ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ↔ 𝑓 ⊆ (On × {1o, 2o}))
9	7, 8	sylibr 236	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑓 ∈ 𝒫 (On × {1o, 2o}))
10		ffun 6679	. . . . . . 7 ⊢ (𝑓:𝑥⟶{1o, 2o} → Fun 𝑓)
11	10	adantl 484	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → Fun 𝑓)
12		fdm 6686	. . . . . . . 8 ⊢ (𝑓:𝑥⟶{1o, 2o} → dom 𝑓 = 𝑥)
13	12	adantl 484	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 = 𝑥)
14		simpl 485	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → 𝑥 ∈ On)
15	13, 14	eqeltrd 2852	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → dom 𝑓 ∈ On)
16	9, 11, 15	jca32 522	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑓:𝑥⟶{1o, 2o}) → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
17	16	rexlimiva 3145	. . . 4 ⊢ (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} → (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
18		simprr 780	. . . . 5 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → dom 𝑓 ∈ On)
19		feq2 6655	. . . . . 6 ⊢ (𝑥 = dom 𝑓 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝑓:dom 𝑓⟶{1o, 2o}))
20	19	adantl 484	. . . . 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 485	. . . . . 6 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → Fun 𝑓)
22		elpwi 4552	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (On × {1o, 2o}) → 𝑓 ⊆ (On × {1o, 2o}))
23		funssxp 6705	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) ↔ (𝑓:dom 𝑓⟶{1o, 2o} ∧ dom 𝑓 ⊆ On))
24	23	simplbi 499	. . . . . 6 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (On × {1o, 2o})) → 𝑓:dom 𝑓⟶{1o, 2o})
25	21, 22, 24	syl2anr 605	. . . . 5 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → 𝑓:dom 𝑓⟶{1o, 2o})
26	18, 20, 25	rspcedvd 3574	. . . 4 ⊢ ((𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)) → ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o})
27	17, 26	impbii 211	. . 3 ⊢ (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On)))
28	27	abbii 2819	. 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} = {𝑓 ∣ (𝑓 ∈ 𝒫 (On × {1o, 2o}) ∧ (Fun 𝑓 ∧ dom 𝑓 ∈ On))}
29		df-no 27673	. 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
30		df-rab 3405	. 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 2785	1 ⊢ No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  ∃wrex 3076  {crab 3404   ⊆ wss 3895  𝒫 cpw 4545  {cpr 4574   × cxp 5634  dom cdm 5636  Oncon0 6331  Fun wfun 6500  ⟶wf 6502  1_oc1o 8414  2_oc2o 8415   No csur 27670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-tr 5198  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-dm 5646  df-rn 5647  df-ord 6334  df-on 6335  df-fun 6508  df-fn 6509  df-f 6510  df-no 27673

This theorem is referenced by: (None)