Theorem sategoelfvb 36270

Description: Characterization of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)

Hypothesis

Ref	Expression
sategoelfvb.s	⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))

Assertion

Ref	Expression
sategoelfvb	⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))

Proof of Theorem sategoelfvb

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sategoelfvb.s	. . . . 5 ⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))
2		ovexd 7455	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) ∈ V)
3		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
4		opeq1 4839	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	opeq2d 4846	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨∅, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
6	5	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
8	7	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑎 = 𝐴) → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
9		simpr 490	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ ω)
10		opeq2 4840	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	opeq2d 4846	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨∅, ⟨𝐴, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
12	11	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
13	12	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑏 = 𝐵) → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
14		eqidd 2761	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3578	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3578	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 36198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 36198	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2774	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3225	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
21	16, 20	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏))
22		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (𝑥 = (𝑎∈𝑔𝑏) ↔ (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
23	22	2rexbidv 3227	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
24		fmla0 36233	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏)}
25	23, 24	elrab2 3648	. . . . . . 7 ⊢ ((𝐴∈𝑔𝐵) ∈ (Fmla‘∅) ↔ ((𝐴∈𝑔𝐵) ∈ V ∧ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
26	2, 21, 25	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) ∈ (Fmla‘∅))
27		satefvfmla0 36269	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴∈𝑔𝐵) ∈ (Fmla‘∅)) → (𝑀 Sat∈ (𝐴∈𝑔𝐵)) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))})
28	26, 27	sylan2 605	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑀 Sat∈ (𝐴∈𝑔𝐵)) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))})
29	1, 28	eqtrid 2807	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → 𝐸 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))})
30	29	eleq2d 2846	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ 𝑆 ∈ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))}))
31		fveq1 6884	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))))
32		fveq1 6884	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))))
33	31, 32	eleq12d 2854	. . . 4 ⊢ (𝑎 = 𝑆 → ((𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) ↔ (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))))
34	33	elrab 3644	. . 3 ⊢ (𝑆 ∈ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))} ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))))
35	30, 34	bitrdi 290	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))))))
36	17	fveq2d 6889	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(𝐴∈𝑔𝐵)) = (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5260	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5431	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 8010	. . . . . . . . 9 ⊢ (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6888	. . . . . . . 8 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1st ‘⟨𝐴, 𝐵⟩)
42		op1stg 8013	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2807	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2795	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐴)
45	44	fveq2d 6889	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6888	. . . . . . . 8 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 8014	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2807	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2795	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐵)
51	50	fveq2d 6889	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2854	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
53	52	adantl 487	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
54	53	anbi2d 642	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵))))) ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))
55	35, 54	bitrd 282	1 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450  ∅c0 4278  ⟨cop 4834  ‘cfv 6538  (class class class)co 7420  ωcom 7877  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8850  ∈_𝑔cgoe 36184  Fmlacfmla 36188   Sat_∈ csate 36189

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-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-ac2 10544

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-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-er 8720  df-map 8852  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-ac 10198  df-goel 36191  df-gona 36192  df-goal 36193  df-sat 36194  df-sate 36195  df-fmla 36196

This theorem is used by:  sategoelfv  36271  ex-sategoelel  36272  ex-sategoelelomsuc  36277  ex-sategoelel12  36278