Theorem ecscottd 35885

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 35884	. . . 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 8758	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
6	2, 5	syl 18	. . . . . . 7 ⊢ (𝜑 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
7	1	ersymb 8770	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝐴))
8	7	abbidv 2829	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ 𝐴𝑅𝑥} = {𝑥 ∣ 𝑥𝑅𝐴})
9	6, 8	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → [𝐴]𝑅 = {𝑥 ∣ 𝑥𝑅𝐴})
10	9	scotteqd 9967	. . . . 5 ⊢ (𝜑 → Scott [𝐴]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐴})
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → Scott [𝐴]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐴})
12		dfec2 8758	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → [𝐵]𝑅 = {𝑥 ∣ 𝐵𝑅𝑥})
13	1	ersymb 8770	. . . . . . 7 ⊢ (𝜑 → (𝐵𝑅𝑥 ↔ 𝑥𝑅𝐵))
14	13	abbidv 2829	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ 𝐵𝑅𝑥} = {𝑥 ∣ 𝑥𝑅𝐵})
15	12, 14	sylan9eqr 2820	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → [𝐵]𝑅 = {𝑥 ∣ 𝑥𝑅𝐵})
16	15	scotteqd 9967	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑉) → Scott [𝐵]𝑅 = Scott {𝑥 ∣ 𝑥𝑅𝐵})
17	11, 16	eqeq12d 2779	. . 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 8769	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ 𝑉)
22	21	stoic1a 1805	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → ¬ 𝐴𝑅𝐵)
23		erdm 8766	. . . . . . . . . 10 ⊢ (𝑅 Er 𝑉 → dom 𝑅 = 𝑉)
24	1, 23	syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 = 𝑉)
25	2, 24	eleqtrrd 2867	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
26		ecdmn0 8808	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
27	25, 26	sylib 221	. . . . . . 7 ⊢ (𝜑 → [𝐴]𝑅 ≠ ∅)
28		scott0b 9974	. . . . . . . 8 ⊢ ([𝐴]𝑅 = ∅ ↔ Scott [𝐴]𝑅 = ∅)
29	28	necon3bii 3011	. . . . . . 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 2850	. . . . . . . . 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 8808	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
36	35	necon2bbii 3010	. . . . . . 7 ⊢ ([𝐵]𝑅 = ∅ ↔ ¬ 𝐵 ∈ dom 𝑅)
37	34, 36	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → [𝐵]𝑅 = ∅)
38		scott0b 9974	. . . . . 6 ⊢ ([𝐵]𝑅 = ∅ ↔ Scott [𝐵]𝑅 = ∅)
39	37, 38	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → Scott [𝐵]𝑅 = ∅)
40	31, 39	neeqtrrd 3033	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑉) → Scott [𝐴]𝑅 ≠ Scott [𝐵]𝑅)
41	40	neneqd 2964	. . 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 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∅c0 4281   class class class wbr 5108  dom cdm 5658   Er wer 8752  [cec 8753  Scott cscott 9965

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-ec 8757  df-r1 9805  df-rank 9806  df-scott 9966

This theorem is used by: (None)