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Theorem nogesgn1ores 27177
Description: Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nogesgn1ores (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

Proof of Theorem nogesgn1ores
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmres 6004 . . . 4 dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2 simp11 1204 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → 𝐴 No )
3 nodmord 27156 . . . . . . 7 (𝐴 No → Ord dom 𝐴)
42, 3syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Ord dom 𝐴)
5 ndmfv 6927 . . . . . . . . . 10 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
6 1n0 8488 . . . . . . . . . . . . 13 1o ≠ ∅
76necomi 2996 . . . . . . . . . . . 12 ∅ ≠ 1o
8 neeq1 3004 . . . . . . . . . . . 12 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ≠ 1o ↔ ∅ ≠ 1o))
97, 8mpbiri 258 . . . . . . . . . . 11 ((𝐴𝑋) = ∅ → (𝐴𝑋) ≠ 1o)
109neneqd 2946 . . . . . . . . . 10 ((𝐴𝑋) = ∅ → ¬ (𝐴𝑋) = 1o)
115, 10syl 17 . . . . . . . . 9 𝑋 ∈ dom 𝐴 → ¬ (𝐴𝑋) = 1o)
1211con4i 114 . . . . . . . 8 ((𝐴𝑋) = 1o𝑋 ∈ dom 𝐴)
1312adantl 483 . . . . . . 7 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) → 𝑋 ∈ dom 𝐴)
14133ad2ant2 1135 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → 𝑋 ∈ dom 𝐴)
15 ordsucss 7806 . . . . . 6 (Ord dom 𝐴 → (𝑋 ∈ dom 𝐴 → suc 𝑋 ⊆ dom 𝐴))
164, 14, 15sylc 65 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → suc 𝑋 ⊆ dom 𝐴)
17 df-ss 3966 . . . . 5 (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
1816, 17sylib 217 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
191, 18eqtrid 2785 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
20 dmres 6004 . . . 4 dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
21 simp12 1205 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → 𝐵 No )
22 nodmord 27156 . . . . . . 7 (𝐵 No → Ord dom 𝐵)
2321, 22syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Ord dom 𝐵)
24 nogesgn1o 27176 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)
25 ndmfv 6927 . . . . . . . . 9 𝑋 ∈ dom 𝐵 → (𝐵𝑋) = ∅)
26 neeq1 3004 . . . . . . . . . . 11 ((𝐵𝑋) = ∅ → ((𝐵𝑋) ≠ 1o ↔ ∅ ≠ 1o))
277, 26mpbiri 258 . . . . . . . . . 10 ((𝐵𝑋) = ∅ → (𝐵𝑋) ≠ 1o)
2827neneqd 2946 . . . . . . . . 9 ((𝐵𝑋) = ∅ → ¬ (𝐵𝑋) = 1o)
2925, 28syl 17 . . . . . . . 8 𝑋 ∈ dom 𝐵 → ¬ (𝐵𝑋) = 1o)
3029con4i 114 . . . . . . 7 ((𝐵𝑋) = 1o𝑋 ∈ dom 𝐵)
3124, 30syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → 𝑋 ∈ dom 𝐵)
32 ordsucss 7806 . . . . . 6 (Ord dom 𝐵 → (𝑋 ∈ dom 𝐵 → suc 𝑋 ⊆ dom 𝐵))
3323, 31, 32sylc 65 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → suc 𝑋 ⊆ dom 𝐵)
34 df-ss 3966 . . . . 5 (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3533, 34sylib 217 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3620, 35eqtrid 2785 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
3719, 36eqtr4d 2776 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
3819eleq2d 2820 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
39 vex 3479 . . . . . . . . 9 𝑥 ∈ V
4039elsuc 6435 . . . . . . . 8 (𝑥 ∈ suc 𝑋 ↔ (𝑥𝑋𝑥 = 𝑋))
41 simpl2l 1227 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → (𝐴𝑋) = (𝐵𝑋))
4241fveq1d 6894 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
43 simpr 486 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → 𝑥𝑋)
4443fvresd 6912 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
4543fvresd 6912 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
4642, 44, 453eqtr3d 2781 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → (𝐴𝑥) = (𝐵𝑥))
4746ex 414 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥𝑋 → (𝐴𝑥) = (𝐵𝑥)))
48 simp2r 1201 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴𝑋) = 1o)
4948, 24eqtr4d 2776 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴𝑋) = (𝐵𝑋))
50 fveq2 6892 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
51 fveq2 6892 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
5250, 51eqeq12d 2749 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑋) = (𝐵𝑋)))
5349, 52syl5ibrcom 246 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 = 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5447, 53jaod 858 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → ((𝑥𝑋𝑥 = 𝑋) → (𝐴𝑥) = (𝐵𝑥)))
5540, 54biimtrid 241 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 ∈ suc 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5655imp 408 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → (𝐴𝑥) = (𝐵𝑥))
57 simpr 486 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → 𝑥 ∈ suc 𝑋)
5857fvresd 6912 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴𝑥))
5957fvresd 6912 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → ((𝐵 ↾ suc 𝑋)‘𝑥) = (𝐵𝑥))
6056, 58, 593eqtr4d 2783 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
6160ex 414 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 ∈ suc 𝑋 → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
6238, 61sylbid 239 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
6362ralrimiv 3146 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ suc 𝑋)((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
64 nofun 27152 . . . . 5 (𝐴 No → Fun 𝐴)
652, 64syl 17 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun 𝐴)
6665funresd 6592 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun (𝐴 ↾ suc 𝑋))
67 nofun 27152 . . . . 5 (𝐵 No → Fun 𝐵)
6821, 67syl 17 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun 𝐵)
6968funresd 6592 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun (𝐵 ↾ suc 𝑋))
70 eqfunfv 7038 . . 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))))
7166, 69, 70syl2anc 585 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → ((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))))
7237, 63, 71mpbir2and 712 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  cin 3948  wss 3949  c0 4323   class class class wbr 5149  dom cdm 5677  cres 5679  Ord word 6364  Oncon0 6365  suc csuc 6367  Fun wfun 6538  cfv 6544  1oc1o 8459   No csur 27143   <s cslt 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147
This theorem is referenced by:  noinfbnd1lem3  27228
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