sategoelfvb - Mathbox for Mario Carneiro

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Theorem sategoelfvb 34479

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 7446	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈_𝑔𝐵) ∈ V)
3		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
4		opeq1 4873	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	opeq2d 4880	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨∅, ⟨𝑎, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
6	5	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
7	6	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
8	7	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑎 = 𝐴) → (∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩ ↔ ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩))
9		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ ω)
10		opeq2 4874	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	opeq2d 4880	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨∅, ⟨𝐴, 𝑏⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
12	11	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
13	12	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑏 = 𝐵) → (⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩ ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩))
14		eqidd 2733	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
15	9, 13, 14	rspcedvd 3614	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝐴, 𝑏⟩⟩)
16	3, 8, 15	rspcedvd 3614	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
17		goel 34407	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈_𝑔𝐵) = ⟨∅, ⟨𝐴, 𝐵⟩⟩)
18		goel 34407	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎∈_𝑔𝑏) = ⟨∅, ⟨𝑎, 𝑏⟩⟩)
19	17, 18	eqeqan12d 2746	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏) ↔ ⟨∅, ⟨𝐴, 𝐵⟩⟩ = ⟨∅, ⟨𝑎, 𝑏⟩⟩))
20	19	2rexbidva 3217	. . . . . . . 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 256	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏))
22		eqeq1 2736	. . . . . . . . 9 ⊢ (𝑥 = (𝐴∈_𝑔𝐵) → (𝑥 = (𝑎∈_𝑔𝑏) ↔ (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏)))
23	22	2rexbidv 3219	. . . . . . . 8 ⊢ (𝑥 = (𝐴∈_𝑔𝐵) → (∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈_𝑔𝑏) ↔ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏)))
24		fmla0 34442	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑎 ∈ ω ∃𝑏 ∈ ω 𝑥 = (𝑎∈_𝑔𝑏)}
25	23, 24	elrab2 3686	. . . . . . 7 ⊢ ((𝐴∈_𝑔𝐵) ∈ (Fmla‘∅) ↔ ((𝐴∈_𝑔𝐵) ∈ V ∧ ∃𝑎 ∈ ω ∃𝑏 ∈ ω (𝐴∈_𝑔𝐵) = (𝑎∈_𝑔𝑏)))
26	2, 21, 25	sylanbrc 583	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴∈_𝑔𝐵) ∈ (Fmla‘∅))
27		satefvfmla0 34478	. . . . . 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 593	. . . . 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 2784	. . . 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 2819	. . 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 6890	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))))
32		fveq1 6890	. . . . 5 ⊢ (𝑎 = 𝑆 → (𝑎‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))))
33	31, 32	eleq12d 2827	. . . 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 3683	. . 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 286	. 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 6895	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(𝐴∈_𝑔𝐵)) = (2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩))
37	36	fveq2d 6895	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
38		0ex 5307	. . . . . . . . . 10 ⊢ ∅ ∈ V
39		opex 5464	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
40	38, 39	op2nd 7986	. . . . . . . . 9 ⊢ (2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩) = ⟨𝐴, 𝐵⟩
41	40	fveq2i 6894	. . . . . . . 8 ⊢ (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (1^st ‘⟨𝐴, 𝐵⟩)
42		op1stg 7989	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43	41, 42	eqtrid 2784	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐴)
44	37, 43	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = 𝐴)
45	44	fveq2d 6895	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(1^st ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘𝐴))
46	36	fveq2d 6895	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)))
47	40	fveq2i 6894	. . . . . . . 8 ⊢ (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = (2^nd ‘⟨𝐴, 𝐵⟩)
48		op2ndg 7990	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	47, 48	eqtrid 2784	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘⟨∅, ⟨𝐴, 𝐵⟩⟩)) = 𝐵)
50	46, 49	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵))) = 𝐵)
51	50	fveq2d 6895	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑆‘(2^nd ‘(2^nd ‘(𝐴∈_𝑔𝐵)))) = (𝑆‘𝐵))
52	45, 51	eleq12d 2827	. . . 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 482	. . 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 629	. 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 278	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 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 {crab 3432 Vcvv 3474 ∅c0 4322 ⟨cop 4634 ‘cfv 6543 (class class class)co 7411 ωcom 7857 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 ∈_𝑔cgoe 34393 Fmlacfmla 34397 Sat_∈ csate 34398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-ac2 10460

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-ac 10113 df-goel 34400 df-gona 34401 df-goal 34402 df-sat 34403 df-sate 34404 df-fmla 34405

This theorem is referenced by: sategoelfv 34480 ex-sategoelel 34481 ex-sategoelelomsuc 34486 ex-sategoelel12 34487

W3C validator