Theorem sategoelfvb 36364

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 7481	. . . . . . 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 2774	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3190	. . . . . . . . . 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 2774	. . . . . . . . . . 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 2764	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3582	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3582	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 36292	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 36292	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2777	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3229	. . . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (𝑥 = (𝑎∈𝑔𝑏) ↔ (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
23	22	2rexbidv 3231	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
24		fmla0 36327	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏)}
25	23, 24	elrab2 3652	. . . . . . 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 36363	. . . . . 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 2810	. . . 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 2850	. . 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 6910	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))))
32		fveq1 6910	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))))
33	31, 32	eleq12d 2858	. . . 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 3648	. . 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 6915	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(𝐴∈𝑔𝐵)) = (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6915	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5265	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5437	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 8037	. . . . . . . . 9 ⊢ (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6914	. . . . . . . 8 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1st ‘⟨𝐴, 𝐵⟩)
42		op1stg 8040	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2810	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2798	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐴)
45	44	fveq2d 6915	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6915	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6914	. . . . . . . 8 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 8041	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2810	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2798	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐵)
51	50	fveq2d 6915	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2858	. . . 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 2147  ∃wrex 3090  {crab 3416  Vcvv 3454  ∅c0 4281  ⟨cop 4838  ‘cfv 6561  (class class class)co 7446  ωcom 7904  1^st c1st 8026  2^nd c2nd 8027   ↑_m cmap 8878  ∈_𝑔cgoe 36278  Fmlacfmla 36282   Sat_∈ csate 36283

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-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-ac2 10571

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-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-card 10050  df-ac 10225  df-goel 36285  df-gona 36286  df-goal 36287  df-sat 36288  df-sate 36289  df-fmla 36290

This theorem is used by:  sategoelfv  36365  ex-sategoelel  36366  ex-sategoelelomsuc  36371  ex-sategoelel12  36372