Theorem rdgeqoa 38485

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 1156	. 2 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → 𝑋 ∈ ω)
2		eleq1 2852	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ω ↔ 𝑋 ∈ ω))
3	2	3anbi3d 1471	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω)))
4		oveq2 7451	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑁 +o 𝑥) = (𝑁 +o 𝑋))
5	4	fveq2d 6912	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)))
6		oveq2 7451	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀 +o 𝑥) = (𝑀 +o 𝑋))
7	6	fveq2d 6912	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))
8	5, 7	eqeq12d 2779	. . . . 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 343	. . . 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 347	. . 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 7924	. . . . 5 ⊢ ∅ ∈ ω
12		oa0 8552	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ On → (𝑁 +o ∅) = 𝑁)
13	12	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐴)‘𝑁))
14	13	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
15		oa0 8552	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On → (𝑀 +o ∅) = 𝑀)
16	15	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘(𝑀 +o ∅)) = (rec(𝐹, 𝐵)‘𝑀))
17	16	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘𝑀) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
18	14, 17	eqeqan12d 2777	. . . . . . . . 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 232	. . . . . . . 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 2852	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ∈ ω ↔ ∅ ∈ ω))
21	20	3anbi3d 1471	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω)))
22	11	biantru 539	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On ↔ (𝑀 ∈ On ∧ ∅ ∈ ω))
23	22	anbi2i 635	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
24		3anass 1111	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
25	23, 24	bitr4i 281	. . . . . . . . . 10 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω))
26	21, 25	bitr4di 292	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On)))
27		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑁 +o 𝑥) = (𝑁 +o ∅))
28	27	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
29		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 +o 𝑥) = (𝑀 +o ∅))
30	29	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
31	28, 30	eqeq12d 2779	. . . . . . . . . 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 343	. . . . . . . . 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 347	. . . . . . . 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 261	. . . . . . 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 1828	. . . . . 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 3772	. . . . . 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 261	. . . . 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 7918	. . . . 5 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
40	39	3anbi3i 1177	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω))
41	40	imbi1i 352	. . . . . . 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 7907	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
43		oacl 8571	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ On) → (𝑁 +o 𝑥) ∈ On)
44		oacl 8571	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ On) → (𝑀 +o 𝑥) ∈ On)
45	43, 44	anim12i 625	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑀 ∈ On ∧ 𝑥 ∈ On)) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
46	45	3impdir 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
47		rdgsuc 8460	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 +o 𝑥) ∈ On → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥))))
48		fveq2 6908	. . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 8460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 +o 𝑥) ∈ On → (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
52	51	ad2antll 742	. . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . 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 464	. . . . . . . . . . . . 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 1442	. . . . . . . . . . . 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 8564	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (𝑁 +o suc 𝑥) = suc (𝑁 +o 𝑥))
58	57	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
59	58	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
60	59	adantr 486	. . . . . . . . . . . 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 8564	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (𝑀 +o suc 𝑥) = suc (𝑀 +o 𝑥))
62	61	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
63	62	3adant1 1148	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
64	63	adantr 486	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 418	. . . . . . . . . 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 58	. . . . . . . . 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 238	. . . . . . . 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 15	. . . . . . 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 220	. . . . . 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 3790	. . . . . . 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 3806	. . . . . . . . 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 3814	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑁 ∈ On ↔ 𝑁 ∈ On))
74		sbcg 3814	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑀 ∈ On ↔ 𝑀 ∈ On))
75		sbcel1v 3807	. . . . . . . . . . 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 1465	. . . . . . . . 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 286	. . . . . . . 8 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
79		sbcimg 3790	. . . . . . . . 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 3814	. . . . . . . . . 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 4373	. . . . . . . . . . 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 3870	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴) = rec(𝐹, 𝐴))
84		csbov123 7487	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥)
85		csbconstg 3870	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌ +o = +o )
86		csbconstg 3870	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑁 = 𝑁)
87		csbvarg 4395	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑥 = suc 𝑥)
88	85, 86, 87	oveq123d 7464	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑁 +o suc 𝑥))
89	84, 88	eqtrid 2810	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑁 +o 𝑥) = (𝑁 +o suc 𝑥))
90	83, 89	fveq12d 6915	. . . . . . . . . . . . 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 2810	. . . . . . . . . . . 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 3870	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵) = rec(𝐹, 𝐵))
94		csbov123 7487	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥)
95		csbconstg 3870	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑀 = 𝑀)
96	85, 95, 87	oveq123d 7464	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +o ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑀 +o suc 𝑥))
97	94, 96	eqtrid 2810	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑀 +o 𝑥) = (𝑀 +o suc 𝑥))
98	93, 97	fveq12d 6915	. . . . . . . . . . . . 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 2810	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
100	91, 99	eqeq12d 2779	. . . . . . . . . . 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 282	. . . . . . . . . 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 347	. . . . . . . . 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 282	. . . . . . . 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 347	. . . . . . 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 282	. . . . . 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 249	. . . . 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 220	. . . 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 7936	. . 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 3540	. 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 39	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147  [wsbc 3742  ⦋csb 3851  ∅c0 4281  Oncon0 6380  suc csuc 6382  ‘cfv 6558  (class class class)co 7443  ωcom 7901  reccrdg 8445   +_o coa 8501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-oadd 8508

This theorem is used by:  finxpreclem4  38509