Theorem tfrlem9 6528

Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem9	⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem9

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldm2g 4933	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹)))
2	1	ibi 176	. 2 ⊢ (𝐵 ∈ dom recs(𝐹) → ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹))
3		df-recs 6514	. . . . . 6 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
4	3	eleq2i 2298	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))})
5		eluniab 3910	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
6	4, 5	bitri 184	. . . 4 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
7		fnop 5442	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵 ∈ 𝑥)
8		rspe 2582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
9		tfrlem.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
10	9	abeq2i 2342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
11		elssuni 3926	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ ∪ 𝐴)
12	9	recsfval 6524	. . . . . . . . . . . . . . . . . 18 ⊢ recs(𝐹) = ∪ 𝐴
13	11, 12	sseqtrrdi 3277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ recs(𝐹))
14	10, 13	sylbir 135	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → 𝑓 ⊆ recs(𝐹))
15	8, 14	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → 𝑓 ⊆ recs(𝐹))
16		fveq2 5648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝑓‘𝑦) = (𝑓‘𝐵))
17		reseq2 5014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐵 → (𝑓 ↾ 𝑦) = (𝑓 ↾ 𝐵))
18	17	fveq2d 5652	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝐵)))
19	16, 18	eqeq12d 2246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
20	19	rspcv 2907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
21		fndm 5436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
22	21	eleq2d 2301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 ↔ 𝐵 ∈ 𝑥))
23	9	tfrlem7 6526	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Fun recs(𝐹)
24		funssfv 5674	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
25	23, 24	mp3an1 1361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
26	25	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
27	21	eleq1d 2300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28		onelss 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓))
29	27, 28	biimtrrdi 164	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓)))
30	29	imp31 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31		fun2ssres 5377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓 ↾ 𝐵))
32	31	fveq2d 5652	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
33	23, 32	mp3an1 1361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
34	30, 33	sylan2 286	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
35	26, 34	eqeq12d 2246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → ((recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
36	35	exbiri 382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ⊆ recs(𝐹) → (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
37	36	com3l 81	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
38	37	exp31 364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
39	38	com34 83	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝐵 ∈ dom 𝑓 → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
40	39	com24 87	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
41	22, 40	sylbird 170	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ 𝑥 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
42	41	com3l 81	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
43	20, 42	syld 45	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
44	43	com24 87	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ 𝑥 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
45	44	imp4d 352	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
46	15, 45	mpdi 43	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
47	7, 46	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
48	47	exp4d 369	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
49	48	ex 115	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
50	49	com4r 86	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
51	50	pm2.43i 49	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
52	51	com3l 81	. . . . . . . 8 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
53	52	imp4a 349	. . . . . . 7 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
54	53	rexlimdv 2650	. . . . . 6 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
55	54	imp 124	. . . . 5 ⊢ ((⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
56	55	exlimiv 1647	. . . 4 ⊢ (∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
57	6, 56	sylbi 121	. . 3 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
58	57	exlimiv 1647	. 2 ⊢ (∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
59	2, 58	syl 14	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512   ⊆ wss 3201  ⟨cop 3676  ∪ cuni 3898  Oncon0 4466  dom cdm 4731   ↾ cres 4733  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333  recscrecs 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfr2a  6530  tfrlemiubacc  6539  tfr1onlemubacc  6555  tfrcllemubacc  6568