Theorem tmachlem-agreeprod 47947

Description: Agreement set can be written as infinite product of acceptable values for tape's cells. (Contributed by Ender Ting, 27-Jul-2026.)

Hypotheses

Ref	Expression
tmach.finalph	⊢ (𝜑 → 𝑈 ∈ Fin)
tmach.exindex	⊢ (𝜑 → 𝐼 ∈ V)
tmach.tapelist	⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
tmach.scanmap	⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
tmach.agreemap	⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
tmach.agreement	⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))

Assertion

Ref	Expression
tmachlem-agreeprod	⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))

Distinct variable groups:   𝑈,𝑎,𝑖,𝑦,𝑧   𝐼,𝑎,𝑖,𝑦,𝑧   𝜑,𝑎,𝑖,𝑦,𝑧   𝑆,𝑎,𝑖,𝑦,𝑧   𝐴,𝑎,𝑖,𝑦,𝑧   𝑇,𝑎,𝑖,𝑦,𝑧

Proof of Theorem tmachlem-agreeprod

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑆‘𝑧) = (𝑆‘𝑎))
2	1	reseq2d 5969	. . . . 5 ⊢ (𝑧 = 𝑎 → (𝑦 ↾ (𝑆‘𝑧)) = (𝑦 ↾ (𝑆‘𝑎)))
3		id 23	. . . . . 6 ⊢ (𝑧 = 𝑎 → 𝑧 = 𝑎)
4	3, 1	reseq12d 5970	. . . . 5 ⊢ (𝑧 = 𝑎 → (𝑧 ↾ (𝑆‘𝑧)) = (𝑎 ↾ (𝑆‘𝑎)))
5	2, 4	eqeq12d 2776	. . . 4 ⊢ (𝑧 = 𝑎 → ((𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧)) ↔ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))))
6	5	rabbidv 3419	. . 3 ⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} = {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
7		tmach.agreemap	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
9		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
10		tmach.finalph	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
11		tmach.exindex	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
12		tmach.tapelist	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
13		tmach.scanmap	. . . . . 6 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
14		tmach.agreement	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
15	10, 11, 12, 13, 7, 14	tmachlem-extapes 47944	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ V)
16	15	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ V)
17		ssrab2 4005	. . . . 5 ⊢ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ⊆ 𝑇
18	17	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ⊆ 𝑇)
19	16, 18	ssexd 5285	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ∈ V)
20	6, 8, 9, 19	fvmptd4 7018	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
21		snfi 9071	. . . . . . . . 9 ⊢ {(𝑎‘𝑖)} ∈ Fin
22	21	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → {(𝑎‘𝑖)} ∈ Fin)
23	10	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → 𝑈 ∈ Fin)
24	22, 23	ifcld 4528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ∈ Fin)
25	24	ralrimiva 3154	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ∈ Fin)
26		ixpssmapg 8956	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ∈ Fin → X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ (∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↑m 𝐼))
27	25, 26	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ (∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↑m 𝐼))
28		ifssun 4499	. . . . . . . 8 ⊢ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ ({(𝑎‘𝑖)} ∪ 𝑈)
29	12	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑎 ∈ 𝑇 ↔ 𝑎 ∈ (𝑈 ↑m 𝐼)))
30	29	biimpd 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ 𝑇 → 𝑎 ∈ (𝑈 ↑m 𝐼)))
31	30	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝑈 ↑m 𝐼))
32		elmapi 8869	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑈 ↑m 𝐼) → 𝑎:𝐼⟶𝑈)
33	31, 32	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎:𝐼⟶𝑈)
34	33	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → (𝑎‘𝑖) ∈ 𝑈)
35	34	snssd 4746	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → {(𝑎‘𝑖)} ⊆ 𝑈)
36		ssequn1 4184	. . . . . . . . 9 ⊢ ({(𝑎‘𝑖)} ⊆ 𝑈 ↔ ({(𝑎‘𝑖)} ∪ 𝑈) = 𝑈)
37	35, 36	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → ({(𝑎‘𝑖)} ∪ 𝑈) = 𝑈)
38	28, 37	sseqtrid 3950	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑖 ∈ 𝐼) → if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ 𝑈)
39	38	iunssd 5008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ 𝑈)
40		mapss 8917	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ ∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ 𝑈) → (∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↑m 𝐼) ⊆ (𝑈 ↑m 𝐼))
41	10, 39, 40	syl2an2r 698	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↑m 𝐼) ⊆ (𝑈 ↑m 𝐼))
42	27, 41	sstrd 3918	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ (𝑈 ↑m 𝐼))
43	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 = (𝑈 ↑m 𝐼))
44	42, 43	sseqtrrd 3945	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ⊆ 𝑇)
45		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → 𝑖 ∈ (𝑆‘𝑎))
46		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)))
47	45, 46	mpd 16	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → (𝑦‘𝑖) = (𝑎‘𝑖))
48		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝑦‘𝑖) ∈ V
49	48	elsn 4598	. . . . . . . . . . . 12 ⊢ ((𝑦‘𝑖) ∈ {(𝑎‘𝑖)} ↔ (𝑦‘𝑖) = (𝑎‘𝑖))
50	47, 49	sylibr 237	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → (𝑦‘𝑖) ∈ {(𝑎‘𝑖)})
51	45	iftrued 4489	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) = {(𝑎‘𝑖)})
52	50, 51	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))) → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))
53	52	ex 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)))
54	53	a1dd 51	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) → (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
55	13	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ (𝒫 𝐼 ∩ Fin))
56	55	elin1d 4101	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ 𝒫 𝐼)
57	56	elpwid 4565	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ⊆ 𝐼)
58	57	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑆‘𝑎) ⊆ 𝐼)
59	58	sselda 3908	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → 𝑖 ∈ 𝐼)
60	59	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → 𝑖 ∈ 𝐼)
61		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)))
62	60, 61	mpd 16	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))
63		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → 𝑖 ∈ (𝑆‘𝑎))
64	63	iftrued 4489	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) = {(𝑎‘𝑖)})
65	62, 64	eleqtrd 2862	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → (𝑦‘𝑖) ∈ {(𝑎‘𝑖)})
66	65	elsnd 4601	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → (𝑦‘𝑖) = (𝑎‘𝑖))
67	66	ex 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)) → (𝑦‘𝑖) = (𝑎‘𝑖)))
68	67	a1dd 51	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)) → (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))))
69	54, 68	impbid 215	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) ↔ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
70	12	eleq2d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝑇 ↔ 𝑦 ∈ (𝑈 ↑m 𝐼)))
71	70	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑦 ∈ 𝑇 → 𝑦 ∈ (𝑈 ↑m 𝐼)))
72	71	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑦 ∈ 𝑇 → 𝑦 ∈ (𝑈 ↑m 𝐼)))
73	72	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (𝑈 ↑m 𝐼))
74		elmapi 8869	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑈 ↑m 𝐼) → 𝑦:𝐼⟶𝑈)
75	73, 74	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦:𝐼⟶𝑈)
76	75	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) → 𝑦:𝐼⟶𝑈)
77	76	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ 𝑈)
78		simplr 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ 𝑖 ∈ 𝐼) → ¬ 𝑖 ∈ (𝑆‘𝑎))
79	78	iffalsed 4492	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ 𝑖 ∈ 𝐼) → if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) = 𝑈)
80	77, 79	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))
81	80	ex 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) → (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)))
82	81	a1d 26	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) → (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
83		simplr 781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → ¬ 𝑖 ∈ (𝑆‘𝑎))
84	83	pm2.21d 122	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) ∧ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))) → (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)))
85	84	ex 418	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)) → (𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖))))
86	82, 85	impbid 215	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) ∧ ¬ 𝑖 ∈ (𝑆‘𝑎)) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) ↔ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
87	69, 86	pm2.61dan 825	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑖 ∈ (𝑆‘𝑎) → (𝑦‘𝑖) = (𝑎‘𝑖)) ↔ (𝑖 ∈ 𝐼 → (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
88	87	ralbidv2 3181	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (∀𝑖 ∈ (𝑆‘𝑎)(𝑦‘𝑖) = (𝑎‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)))
89	71	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (𝑈 ↑m 𝐼))
90		elmapfn 8887	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑈 ↑m 𝐼) → 𝑦 Fn 𝐼)
91	89, 90	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 Fn 𝐼)
92	91	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 Fn 𝐼)
93	92	biantrurd 542	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
94	88, 93	bitr2d 283	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑦 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)) ↔ ∀𝑖 ∈ (𝑆‘𝑎)(𝑦‘𝑖) = (𝑎‘𝑖)))
95		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
96	95	elixp 8932	. . . . 5 ⊢ (𝑦 ∈ X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈)))
97	96	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑦 ∈ X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))))
98		elmapfn 8887	. . . . . . 7 ⊢ (𝑎 ∈ (𝑈 ↑m 𝐼) → 𝑎 Fn 𝐼)
99	31, 98	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 Fn 𝐼)
100	99	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑎 Fn 𝐼)
101		fvreseq 7039	. . . . 5 ⊢ (((𝑦 Fn 𝐼 ∧ 𝑎 Fn 𝐼) ∧ (𝑆‘𝑎) ⊆ 𝐼) → ((𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)) ↔ ∀𝑖 ∈ (𝑆‘𝑎)(𝑦‘𝑖) = (𝑎‘𝑖)))
102	92, 100, 58, 101	syl21anc 851	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)) ↔ ∀𝑖 ∈ (𝑆‘𝑎)(𝑦‘𝑖) = (𝑎‘𝑖)))
103	94, 97, 102	3bitr4d 314	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑦 ∈ X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) ↔ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))))
104	44, 103	eqrrabd 4011	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈) = {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
105	20, 104	eqtr4d 2798	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ifcif 4481  𝒫 cpw 4556  {csn 4583  ∪ ciun 4950   ↦ cmpt 5185   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Xcixp 8925  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-1o 8476  df-map 8849  df-ixp 8926  df-en 8974  df-fin 8977

This theorem is used by:  tmachlem-tpopen2  47952