Theorem sategoelfvb 33426

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 7342	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) ∈ V)
3		simpl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
4		opeq1 4809	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	opeq2d 4816	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨∅, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
6	5	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3172	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
8	7	adantl 483	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑎 = 𝐴) → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
9		simpr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ ω)
10		opeq2 4810	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	opeq2d 4816	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨∅, ⟨𝐴, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
12	11	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
13	12	adantl 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑏 = 𝐵) → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
14		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3568	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3568	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 33354	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 33354	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2750	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3208	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏))
22		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (𝑥 = (𝑎∈𝑔𝑏) ↔ (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
23	22	2rexbidv 3210	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
24		fmla0 33389	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈𝑔𝑏)}
25	23, 24	elrab2 3632	. . . . . . 7 ⊢ ((𝐴∈𝑔𝐵) ∈ (Fmla‘∅) ↔ ((𝐴∈𝑔𝐵) ∈ V ∧ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈𝑔𝐵) = (𝑎∈𝑔𝑏)))
26	2, 21, 25	sylanbrc 584	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈𝑔𝐵) ∈ (Fmla‘∅))
27		satefvfmla0 33425	. . . . . 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 594	. . . . 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 2788	. . . 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 2822	. . 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 6803	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))))
32		fveq1 6803	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))))
33	31, 32	eleq12d 2831	. . . 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 3629	. . 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 287	. 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 6808	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(𝐴∈𝑔𝐵)) = (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6808	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5240	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5392	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 7872	. . . . . . . . 9 ⊢ (2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6807	. . . . . . . 8 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1st ‘⟨𝐴, 𝐵⟩)
42		op1stg 7875	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2788	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2776	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1st ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐴)
45	44	fveq2d 6808	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6808	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6807	. . . . . . . 8 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 7876	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2788	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2776	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2nd ‘(2nd ‘(𝐴∈𝑔𝐵))) = 𝐵)
51	50	fveq2d 6808	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2831	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
53	52	adantl 483	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝑆‘(1st ‘(2nd ‘(𝐴∈𝑔𝐵)))) ∈ (𝑆‘(2nd ‘(2nd ‘(𝐴∈𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
54	53	anbi2d 630	. 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 279	1 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∃wrex 3071  {crab 3284  Vcvv 3437  ∅c0 4262  ⟨cop 4571  ‘cfv 6458  (class class class)co 7307  ωcom 7744  1^st c1st 7861  2^nd c2nd 7862   ↑_m cmap 8646  ∈_𝑔cgoe 33340  Fmlacfmla 33344   Sat_∈ csate 33345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9443  ax-ac2 10265

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-2o 8329  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-card 9741  df-ac 9918  df-goel 33347  df-gona 33348  df-goal 33349  df-sat 33350  df-sate 33351  df-fmla 33352

This theorem is referenced by:  sategoelfv  33427  ex-sategoelel  33428  ex-sategoelelomsuc  33433  ex-sategoelel12  33434