Theorem nolesgn2ores 27658

Description: Given 𝐴 less-than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2_o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolesgn2ores	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

Proof of Theorem nolesgn2ores

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmres 5971	. . . 4 ⊢ dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2		simp11 1211	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐴 ∈ No )
3		nodmord 27639	. . . . . . 7 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Ord dom 𝐴)
5		ndmfv 6863	. . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
6		2on 8412	. . . . . . . . . . . . . . 15 ⊢ 2o ∈ On
7	6	elexi 3455	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
8	7	prid2 4698	. . . . . . . . . . . . 13 ⊢ 2o ∈ {1o, 2o}
9	8	nosgnn0i 27645	. . . . . . . . . . . 12 ⊢ ∅ ≠ 2o
10		neeq1 2998	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑋) = ∅ → ((𝐴‘𝑋) ≠ 2o ↔ ∅ ≠ 2o))
11	9, 10	mpbiri 260	. . . . . . . . . . 11 ⊢ ((𝐴‘𝑋) = ∅ → (𝐴‘𝑋) ≠ 2o)
12	11	neneqd 2941	. . . . . . . . . 10 ⊢ ((𝐴‘𝑋) = ∅ → ¬ (𝐴‘𝑋) = 2o)
13	5, 12	syl 17	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐴 → ¬ (𝐴‘𝑋) = 2o)
14	13	con4i 114	. . . . . . . 8 ⊢ ((𝐴‘𝑋) = 2o → 𝑋 ∈ dom 𝐴)
15	14	adantl 483	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) → 𝑋 ∈ dom 𝐴)
16	15	3ad2ant2 1141	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐴)
17		ordsucss 7762	. . . . . 6 ⊢ (Ord dom 𝐴 → (𝑋 ∈ dom 𝐴 → suc 𝑋 ⊆ dom 𝐴))
18	4, 16, 17	sylc 65	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → suc 𝑋 ⊆ dom 𝐴)
19		dfss2 3903	. . . . 5 ⊢ (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
20	18, 19	sylib 220	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
21	1, 20	eqtrid 2788	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
22		dmres 5971	. . . 4 ⊢ dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
23		simp12 1212	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐵 ∈ No )
24		nodmord 27639	. . . . . . 7 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
25	23, 24	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Ord dom 𝐵)
26		nolesgn2o 27657	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2o)
27		ndmfv 6863	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐵 → (𝐵‘𝑋) = ∅)
28		neeq1 2998	. . . . . . . . . . 11 ⊢ ((𝐵‘𝑋) = ∅ → ((𝐵‘𝑋) ≠ 2o ↔ ∅ ≠ 2o))
29	9, 28	mpbiri 260	. . . . . . . . . 10 ⊢ ((𝐵‘𝑋) = ∅ → (𝐵‘𝑋) ≠ 2o)
30	29	neneqd 2941	. . . . . . . . 9 ⊢ ((𝐵‘𝑋) = ∅ → ¬ (𝐵‘𝑋) = 2o)
31	27, 30	syl 17	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom 𝐵 → ¬ (𝐵‘𝑋) = 2o)
32	31	con4i 114	. . . . . . 7 ⊢ ((𝐵‘𝑋) = 2o → 𝑋 ∈ dom 𝐵)
33	26, 32	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐵)
34		ordsucss 7762	. . . . . 6 ⊢ (Ord dom 𝐵 → (𝑋 ∈ dom 𝐵 → suc 𝑋 ⊆ dom 𝐵))
35	25, 33, 34	sylc 65	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → suc 𝑋 ⊆ dom 𝐵)
36		dfss2 3903	. . . . 5 ⊢ (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
37	35, 36	sylib 220	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
38	22, 37	eqtrid 2788	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
39	21, 38	eqtr4d 2779	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
40	21	eleq2d 2827	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
41		vex 3437	. . . . . . . . 9 ⊢ 𝑥 ∈ V
42	41	elsuc 6386	. . . . . . . 8 ⊢ (𝑥 ∈ suc 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋))
43		simp2l 1207	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
44	43	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝐴 ↾ 𝑋)‘𝑥) = ((𝐵 ↾ 𝑋)‘𝑥))
45	44	adantr 482	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑥) = ((𝐵 ↾ 𝑋)‘𝑥))
46		simpr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
47	46	fvresd 6851	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑥) = (𝐴‘𝑥))
48	46	fvresd 6851	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑥) = (𝐵‘𝑥))
49	45, 47, 48	3eqtr3d 2784	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝐴‘𝑥) = (𝐵‘𝑥))
50	49	ex 414	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ 𝑋 → (𝐴‘𝑥) = (𝐵‘𝑥)))
51		simp2r 1208	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴‘𝑋) = 2o)
52	51, 26	eqtr4d 2779	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴‘𝑋) = (𝐵‘𝑋))
53		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
54		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
55	53, 54	eqeq12d 2757	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑋) = (𝐵‘𝑋)))
56	52, 55	syl5ibrcom 249	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐵‘𝑥)))
57	50, 56	jaod 866	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐵‘𝑥)))
58	42, 57	biimtrid 244	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → (𝐴‘𝑥) = (𝐵‘𝑥)))
59	58	imp 408	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → (𝐴‘𝑥) = (𝐵‘𝑥))
60		simpr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → 𝑥 ∈ suc 𝑋)
61	60	fvresd 6851	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴‘𝑥))
62	60	fvresd 6851	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐵 ↾ suc 𝑋)‘𝑥) = (𝐵‘𝑥))
63	59, 61, 62	3eqtr4d 2786	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
64	63	ex 414	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
65	40, 64	sylbid 242	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
66	65	ralrimiv 3132	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ∀𝑥 ∈ dom (𝐴 ↾ suc 𝑋)((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
67		nofun 27635	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
68		funres 6531	. . . 4 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ suc 𝑋))
69	2, 67, 68	3syl 18	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Fun (𝐴 ↾ suc 𝑋))
70		nofun 27635	. . . 4 ⊢ (𝐵 ∈ No → Fun 𝐵)
71		funres 6531	. . . 4 ⊢ (Fun 𝐵 → Fun (𝐵 ↾ suc 𝑋))
72	23, 70, 71	3syl 18	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Fun (𝐵 ↾ suc 𝑋))
73		eqfunfv 6981	. . 3 ⊢ ((Fun (𝐴 ↾ suc 𝑋) ∧ Fun (𝐵 ↾ suc 𝑋)) → ((𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋) ↔ (dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋) ∧ ∀𝑥 ∈ dom (𝐴 ↾ suc 𝑋)((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))))
74	69, 72, 73	syl2anc 591	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋) ↔ (dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋) ∧ ∀𝑥 ∈ dom (𝐴 ↾ suc 𝑋)((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))))
75	39, 66, 74	mpbir2and 720	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264   class class class wbr 5075  dom cdm 5621   ↾ cres 5623  Ord word 6313  Oncon0 6314  suc csuc 6316  Fun wfun 6483  ‘cfv 6489  1_oc1o 8392  2_oc2o 8393   No csur 27625   <s clts 27626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-1o 8399  df-2o 8400  df-no 27628  df-lts 27629

This theorem is referenced by:  nosupbnd1lem3  27696