Theorem ecscottd 35820

Description: Applying the Scott operation to equivalence classes preserves their equality property. (Contributed by BTernaryTau, 20-Oct-2026.)

Hypotheses

Ref	Expression
ecscottd.1	⊢ (𝜑 → 𝑅 Er 𝑉)
ecscottd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
ecscottd	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ Scott [𝐴]𝑅 = Scott [𝐵]𝑅))

Proof of Theorem ecscottd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecscottd.1	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝑉)
2		ecscottd.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	1, 2	erscottd 35819	. . . 4 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))
5		dfec2 8728	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
6	2, 5	syl 18	. . . . . . 7 ⊢ (𝜑 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
7	1	ersymb 8740	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝐴))
8	7	abbidv 2826	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ 𝐴𝑅𝑥} = {𝑥 ∣ 𝑥𝑅𝐴})
9	6, 8	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → [𝐴]𝑅 = {𝑥 ∣ 𝑥𝑅𝐴})
10	9	scotteqd 9937	. . . . 5 ⊢ (𝜑 → Scott [𝐴]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐴})
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → Scott [𝐴]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐴})
12		dfec2 8728	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → [𝐵]𝑅 = {𝑥 ∣ 𝐵𝑅𝑥})
13	1	ersymb 8740	. . . . . . 7 ⊢ (𝜑 → (𝐵𝑅𝑥 ↔ 𝑥𝑅𝐵))
14	13	abbidv 2826	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ 𝐵𝑅𝑥} = {𝑥 ∣ 𝑥𝑅𝐵})
15	12, 14	sylan9eqr 2817	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → [𝐵]𝑅 = {𝑥 ∣ 𝑥𝑅𝐵})
16	15	scotteqd 9937	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → Scott [𝐵]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐵})
17	11, 16	eqeq12d 2776	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → (Scott [𝐴]𝑅 = Scott [𝐵]𝑅 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))
18	4, 17	bitr4d 285	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 ↔ Scott [𝐴]𝑅 = Scott [𝐵]𝑅))
19	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝑅 Er 𝑉)
20		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
21	19, 20	ercl2 8739	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ 𝑉)
22	21	stoic1a 1804	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → ¬ 𝐴𝑅𝐵)
23		erdm 8736	. . . . . . . . . 10 ⊢ (𝑅 Er 𝑉 → dom 𝑅 = 𝑉)
24	1, 23	syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 = 𝑉)
25	2, 24	eleqtrrd 2863	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
26		ecdmn0 8778	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
27	25, 26	sylib 221	. . . . . . 7 ⊢ (𝜑 → [𝐴]𝑅 ≠ ∅)
28		scott0b 9944	. . . . . . . 8 ⊢ ([𝐴]𝑅 = ∅ ↔ Scott [𝐴]𝑅 = ∅)
29	28	necon3bii 3007	. . . . . . 7 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ Scott [𝐴]𝑅 ≠ ∅)
30	27, 29	sylib 221	. . . . . 6 ⊢ (𝜑 → Scott [𝐴]𝑅 ≠ ∅)
31	30	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → Scott [𝐴]𝑅 ≠ ∅)
32	24	eleq2d 2846	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑉))
33	32	notbid 321	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐵 ∈ dom 𝑅 ↔ ¬ 𝐵 ∈ 𝑉))
34	33	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → ¬ 𝐵 ∈ dom 𝑅)
35		ecdmn0 8778	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
36	35	necon2bbii 3006	. . . . . . 7 ⊢ ([𝐵]𝑅 = ∅ ↔ ¬ 𝐵 ∈ dom 𝑅)
37	34, 36	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → [𝐵]𝑅 = ∅)
38		scott0b 9944	. . . . . 6 ⊢ ([𝐵]𝑅 = ∅ ↔ Scott [𝐵]𝑅 = ∅)
39	37, 38	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → Scott [𝐵]𝑅 = ∅)
40	31, 39	neeqtrrd 3029	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → Scott [𝐴]𝑅 ≠ Scott [𝐵]𝑅)
41	40	neneqd 2960	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → ¬ Scott [𝐴]𝑅 = Scott [𝐵]𝑅)
42	22, 41	2falsed 379	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 ↔ Scott [𝐴]𝑅 = Scott [𝐵]𝑅))
43	18, 42	pm2.61dan 825	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ Scott [𝐴]𝑅 = Scott [𝐵]𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∅c0 4277   class class class wbr 5102  dom cdm 5651   Er wer 8722  [cec 8723  Scott cscott 9935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-er 8725  df-ec 8727  df-r1 9775  df-rank 9776  df-scott 9936

This theorem is used by: (None)