Theorem rdgeqoa 37372

Description: If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)

Assertion

Ref	Expression
rdgeqoa	⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))

Proof of Theorem rdgeqoa

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1138	. 2 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → 𝑋 ∈ ω)
2		eleq1 2828	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ω ↔ 𝑋 ∈ ω))
3	2	3anbi3d 1443	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω)))
4		oveq2 7440	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑁 +o 𝑥) = (𝑁 +o 𝑋))
5	4	fveq2d 6909	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)))
6		oveq2 7440	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀 +o 𝑥) = (𝑀 +o 𝑋))
7	6	fveq2d 6909	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))
8	5, 7	eqeq12d 2752	. . . . 5 ⊢ (𝑥 = 𝑋 → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))
9	8	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑋 → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))))
10	3, 9	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑋 → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))))
11		peano1 7911	. . . . 5 ⊢ ∅ ∈ ω
12		oa0 8555	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ On → (𝑁 +o ∅) = 𝑁)
13	12	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐴)‘𝑁))
14	13	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
15		oa0 8555	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On → (𝑀 +o ∅) = 𝑀)
16	15	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘(𝑀 +o ∅)) = (rec(𝐹, 𝐵)‘𝑀))
17	16	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘𝑀) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
18	14, 17	eqeqan12d 2750	. . . . . . . . 9 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
19	18	biimpd 229	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
20		eleq1 2828	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ∈ ω ↔ ∅ ∈ ω))
21	20	3anbi3d 1443	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω)))
22	11	biantru 529	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On ↔ (𝑀 ∈ On ∧ ∅ ∈ ω))
23	22	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
24		3anass 1094	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
25	23, 24	bitr4i 278	. . . . . . . . . 10 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω))
26	21, 25	bitr4di 289	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On)))
27		oveq2 7440	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑁 +o 𝑥) = (𝑁 +o ∅))
28	27	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
29		oveq2 7440	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 +o 𝑥) = (𝑀 +o ∅))
30	29	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
31	28, 30	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
32	31	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))))
33	26, 32	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = ∅ → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))))
34	19, 33	mpbiri 258	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
35	34	ax-gen 1794	. . . . . 6 ⊢ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
36		sbc6g 3817	. . . . . 6 ⊢ (∅ ∈ ω → ([∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))))
37	35, 36	mpbiri 258	. . . . 5 ⊢ (∅ ∈ ω → [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
38	11, 37	ax-mp 5	. . . 4 ⊢ [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
39		peano2b 7905	. . . . 5 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
40	39	3anbi3i 1159	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω))
41	40	imbi1i 349	. . . . . . 7 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
42		nnon 7894	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
43		oacl 8574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ On) → (𝑁 +o 𝑥) ∈ On)
44		oacl 8574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ On) → (𝑀 +o 𝑥) ∈ On)
45	43, 44	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑀 ∈ On ∧ 𝑥 ∈ On)) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
46	45	3impdir 1351	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
47		rdgsuc 8465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 +o 𝑥) ∈ On → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥))))
48		fveq2 6905	. . . . . . . . . . . . . . . . . 18 ⊢ ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) → (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥))) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
49	47, 48	sylan9eqr 2798	. . . . . . . . . . . . . . . . 17 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ (𝑁 +o 𝑥) ∈ On) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
50	49	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
51		rdgsuc 8465	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 +o 𝑥) ∈ On → (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
52	51	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
53	50, 52	eqtr4d 2779	. . . . . . . . . . . . . . 15 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
54	46, 53	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
55	54	ancoms 458	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
56	42, 55	syl3anl3 1415	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
57		onasuc 8567	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (𝑁 +o suc 𝑥) = suc (𝑁 +o 𝑥))
58	57	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
59	58	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
60	59	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
61		onasuc 8567	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (𝑀 +o suc 𝑥) = suc (𝑀 +o 𝑥))
62	61	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
63	62	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
64	63	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
65	56, 60, 64	3eqtr4d 2786	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
66	65	ex 412	. . . . . . . . . 10 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
67	66	imim2d 57	. . . . . . . . 9 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
68	40, 67	sylbir 235	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
69	68	a2i 14	. . . . . . 7 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
70	41, 69	sylbi 217	. . . . . 6 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
71		sbcimg 3836	. . . . . . 7 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
72		sbc3an 3854	. . . . . . . . 9 ⊢ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ ([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω))
73		sbcg 3862	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑁 ∈ On ↔ 𝑁 ∈ On))
74		sbcg 3862	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑀 ∈ On ↔ 𝑀 ∈ On))
75		sbcel1v 3855	. . . . . . . . . . 11 ⊢ ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
76	75	a1i 11	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω))
77	73, 74, 76	3anbi123d 1437	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
78	72, 77	bitrid 283	. . . . . . . 8 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
79		sbcimg 3836	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
80		sbcg 3862	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀)))
81		sbceqg 4411	. . . . . . . . . . 11 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
82		csbfv12 6953	. . . . . . . . . . . . 13 ⊢ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴)‘⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥))
83		csbconstg 3917	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴) = rec(𝐹, 𝐴))
84		csbov123 7476	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥)
85		csbconstg 3917	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌ +o = +o )
86		csbconstg 3917	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑁 = 𝑁)
87		csbvarg 4433	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑥 = suc 𝑥)
88	85, 86, 87	oveq123d 7453	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑁 +o suc 𝑥))
89	84, 88	eqtrid 2788	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥) = (𝑁 +o suc 𝑥))
90	83, 89	fveq12d 6912	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴)‘⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)))
91	82, 90	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)))
92		csbfv12 6953	. . . . . . . . . . . . 13 ⊢ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵)‘⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥))
93		csbconstg 3917	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵) = rec(𝐹, 𝐵))
94		csbov123 7476	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥)
95		csbconstg 3917	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑀 = 𝑀)
96	85, 95, 87	oveq123d 7453	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑀 +o suc 𝑥))
97	94, 96	eqtrid 2788	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥) = (𝑀 +o suc 𝑥))
98	93, 97	fveq12d 6912	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵)‘⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
99	92, 98	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
100	91, 99	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
101	81, 100	bitrd 279	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
102	80, 101	imbi12d 344	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
103	79, 102	bitrd 279	. . . . . . . 8 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
104	78, 103	imbi12d 344	. . . . . . 7 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))))
105	71, 104	bitrd 279	. . . . . 6 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))))
106	70, 105	imbitrrid 246	. . . . 5 ⊢ (suc 𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
107	39, 106	sylbi 217	. . . 4 ⊢ (𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
108	38, 107	findes 7923	. . 3 ⊢ (𝑥 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
109	10, 108	vtoclga 3576	. 2 ⊢ (𝑋 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))))
110	1, 109	mpcom 38	1 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2107  [wsbc 3787  ⦋csb 3898  ∅c0 4332  Oncon0 6383  suc csuc 6385  ‘cfv 6560  (class class class)co 7432  ωcom 7888  reccrdg 8450   +_o coa 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511

This theorem is referenced by:  finxpreclem4  37396