Theorem nolesgn2ores 33802

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 5902	. . . 4 ⊢ dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2		simp11 1201	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐴 ∈ No )
3		nodmord 33783	. . . . . . 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 6786	. . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
6		2on 8275	. . . . . . . . . . . . . . 15 ⊢ 2o ∈ On
7	6	elexi 3441	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
8	7	prid2 4696	. . . . . . . . . . . . 13 ⊢ 2o ∈ {1o, 2o}
9	8	nosgnn0i 33789	. . . . . . . . . . . 12 ⊢ ∅ ≠ 2o
10		neeq1 3005	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑋) = ∅ → ((𝐴‘𝑋) ≠ 2o ↔ ∅ ≠ 2o))
11	9, 10	mpbiri 257	. . . . . . . . . . 11 ⊢ ((𝐴‘𝑋) = ∅ → (𝐴‘𝑋) ≠ 2o)
12	11	neneqd 2947	. . . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) → 𝑋 ∈ dom 𝐴)
16	15	3ad2ant2 1132	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐴)
17		ordsucss 7640	. . . . . 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		df-ss 3900	. . . . 5 ⊢ (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
20	18, 19	sylib 217	. . . 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	syl5eq 2791	. . 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 5902	. . . 4 ⊢ dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
23		simp12 1202	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐵 ∈ No )
24		nodmord 33783	. . . . . . 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 33801	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2o)
27		ndmfv 6786	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐵 → (𝐵‘𝑋) = ∅)
28		neeq1 3005	. . . . . . . . . . 11 ⊢ ((𝐵‘𝑋) = ∅ → ((𝐵‘𝑋) ≠ 2o ↔ ∅ ≠ 2o))
29	9, 28	mpbiri 257	. . . . . . . . . 10 ⊢ ((𝐵‘𝑋) = ∅ → (𝐵‘𝑋) ≠ 2o)
30	29	neneqd 2947	. . . . . . . . 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 7640	. . . . . 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		df-ss 3900	. . . . 5 ⊢ (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
37	35, 36	sylib 217	. . . 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	syl5eq 2791	. . 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 2781	. 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 2824	. . . 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 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
42	41	elsuc 6320	. . . . . . . 8 ⊢ (𝑥 ∈ suc 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋))
43		simp2l 1197	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
44	43	fveq1d 6758	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
47	46	fvresd 6776	. . . . . . . . . . 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 6776	. . . . . . . . . . 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 2786	. . . . . . . . . 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 412	. . . . . . . . 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 1198	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴‘𝑋) = 2o)
52	51, 26	eqtr4d 2781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴‘𝑋) = (𝐵‘𝑋))
53		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
54		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
55	53, 54	eqeq12d 2754	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑋) = (𝐵‘𝑋)))
56	52, 55	syl5ibrcom 246	. . . . . . . . 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 855	. . . . . . . 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	syl5bi 241	. . . . . . 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 406	. . . . . 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 484	. . . . . . 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 6776	. . . . . 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 6776	. . . . . 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 2788	. . . . 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 412	. . . 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 239	. . 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 3106	. 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 33779	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
68		funres 6460	. . . 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 33779	. . . 4 ⊢ (𝐵 ∈ No → Fun 𝐵)
71		funres 6460	. . . 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 6896	. . 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 583	. 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 709	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 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  Ord word 6250  Oncon0 6251  suc csuc 6253  Fun wfun 6412  ‘cfv 6418  1_oc1o 8260  2_oc2o 8261   No csur 33770   <s cslt 33771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774

This theorem is referenced by:  nosupbnd1lem3  33840