Theorem sategoelfvb 35932

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 7458	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) ∈ V)
3		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
4		opeq1 4843	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	opeq2d 4850	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨∅, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
6	5	eqeq2d 2777	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3192	. . . . . . . . . 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 4844	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	opeq2d 4850	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨∅, ⟨𝐴, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
12	11	eqeq2d 2777	. . . . . . . . . . 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 2767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3586	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 35860	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 35860	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3231	. . . . . . . 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 2770	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (𝑥 = (𝑎∈𝑔𝑏) ↔ (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
23	22	2rexbidv 3233	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
24		fmla0 35895	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏)}
25	23, 24	elrab2 3657	. . . . . . 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 35931	. . . . . 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 2813	. . . 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 2852	. . 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 6887	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))))
32		fveq1 6887	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))))
33	31, 32	eleq12d 2860	. . . 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 3653	. . 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 6892	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(𝐴∈𝑔𝐵)) = (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6892	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5275	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5450	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 8004	. . . . . . . . 9 ⊢ (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6891	. . . . . . . 8 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1st ‘⟨𝐴, 𝐵⟩)
42		op1stg 8007	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2813	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2801	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐴)
45	44	fveq2d 6892	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6892	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6891	. . . . . . . 8 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 8008	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2813	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2801	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐵)
51	50	fveq2d 6892	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2860	. . . 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 1570   ∈ wcel 2146  ∃wrex 3092  {crab 3419  Vcvv 3458  ∅c0 4289  ⟨cop 4600  ‘cfv 6543  (class class class)co 7423  ωcom 7871  1^st c1st 7993  2^nd c2nd 7994   ↑_m cmap 8833  ∈_𝑔cgoe 35846  Fmlacfmla 35850   Sat_∈ csate 35851

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-ac2 10465

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-ac 10119  df-goel 35853  df-gona 35854  df-goal 35855  df-sat 35856  df-sate 35857  df-fmla 35858

This theorem is used by:  sategoelfv  35933  ex-sategoelel  35934  ex-sategoelelomsuc  35939  ex-sategoelel12  35940