sategoelfvb - Mathbox for Mario Carneiro

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Theorem sategoelfvb 34410

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 7444	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈_𝑔𝐵) ∈ V)
3		simpl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
4		opeq1 4874	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	opeq2d 4881	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨∅, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
6	5	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3179	. . . . . . . . . 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 4875	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	opeq2d 4881	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨∅, ⟨𝐴, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
12	11	eqeq2d 2744	. . . . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3615	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3615	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 34338	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈_𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 34338	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈_𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2747	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3218	. . . . . . . 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 2737	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈_𝑔𝐵) → (𝑥 = (𝑎∈_𝑔𝑏) ↔ (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏)))
23	22	2rexbidv 3220	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈_𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈_𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏)))
24		fmla0 34373	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈_𝑔𝑏)}
25	23, 24	elrab2 3687	. . . . . . 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 34409	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴∈_𝑔𝐵) ∈ (Fmla‘∅)) → (𝑀 Sat_∈ (𝐴∈_𝑔𝐵)) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))})
28	26, 27	sylan2 594	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑀 Sat_∈ (𝐴∈_𝑔𝐵)) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))})
29	1, 28	eqtrid 2785	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → 𝐸 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))})
30	29	eleq2d 2820	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ 𝑆 ∈ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))}))
31		fveq1 6891	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))))
32		fveq1 6891	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))))
33	31, 32	eleq12d 2828	. . . 4 ⊢ (𝑎 = 𝑆 → ((𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ↔ (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))))
34	33	elrab 3684	. . 3 ⊢ (𝑆 ∈ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))} ↔ (𝑆 ∈ (𝑀 ↑_m ω) ∧ (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))))))
35	30, 34	bitrdi 287	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑_m ω) ∧ (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))))))
36	17	fveq2d 6896	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(𝐴∈_𝑔𝐵)) = (2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6896	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5308	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5465	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 7984	. . . . . . . . 9 ⊢ (2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6895	. . . . . . . 8 ⊢ (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1^st ‘⟨𝐴, 𝐵⟩)
42		op1stg 7987	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2785	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2773	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = 𝐴)
45	44	fveq2d 6896	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6896	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6895	. . . . . . . 8 ⊢ (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2^nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 7988	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2785	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2773	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = 𝐵)
51	50	fveq2d 6896	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2828	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
53	52	adantl 483	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ↔ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))
54	53	anbi2d 630	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝑆 ∈ (𝑀 ↑_m ω) ∧ (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) ∈ (𝑆‘(2^nd ‘(2^nd ‘(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))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 Vcvv 3475 ∅c0 4323 ⟨cop 4635 ‘cfv 6544 (class class class)co 7409 ωcom 7855 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 ∈_𝑔cgoe 34324 Fmlacfmla 34328 Sat_∈ csate 34329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-ac 10111 df-goel 34331 df-gona 34332 df-goal 34333 df-sat 34334 df-sate 34335 df-fmla 34336

This theorem is referenced by: sategoelfv 34411 ex-sategoelel 34412 ex-sategoelelomsuc 34417 ex-sategoelel12 34418

W3C validator