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Theorem nogesgn1ores 27038
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 5964 . . . 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 27017 . . . . . . 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 6882 . . . . . . . . . 10 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
6 1n0 8439 . . . . . . . . . . . . 13 1o ≠ ∅
76necomi 2999 . . . . . . . . . . . 12 ∅ ≠ 1o
8 neeq1 3007 . . . . . . . . . . . 12 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ≠ 1o ↔ ∅ ≠ 1o))
97, 8mpbiri 258 . . . . . . . . . . 11 ((𝐴𝑋) = ∅ → (𝐴𝑋) ≠ 1o)
109neneqd 2949 . . . . . . . . . 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 7758 . . . . . 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 3932 . . . . 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 2789 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
20 dmres 5964 . . . 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 27017 . . . . . . 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 27037 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)
25 ndmfv 6882 . . . . . . . . 9 𝑋 ∈ dom 𝐵 → (𝐵𝑋) = ∅)
26 neeq1 3007 . . . . . . . . . . 11 ((𝐵𝑋) = ∅ → ((𝐵𝑋) ≠ 1o ↔ ∅ ≠ 1o))
277, 26mpbiri 258 . . . . . . . . . 10 ((𝐵𝑋) = ∅ → (𝐵𝑋) ≠ 1o)
2827neneqd 2949 . . . . . . . . 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 7758 . . . . . 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 3932 . . . . 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 2789 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
3719, 36eqtr4d 2780 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
3819eleq2d 2824 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
39 vex 3452 . . . . . . . . 9 𝑥 ∈ V
4039elsuc 6392 . . . . . . . 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 6849 . . . . . . . . . . 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 6867 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
4543fvresd 6867 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥𝑋) → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
4642, 44, 453eqtr3d 2785 . . . . . . . . . 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 2780 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴𝑋) = (𝐵𝑋))
50 fveq2 6847 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
51 fveq2 6847 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
5250, 51eqeq12d 2753 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑋) = (𝐵𝑋)))
5349, 52syl5ibrcom 247 . . . . . . . . 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 6867 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴𝑥))
5957fvresd 6867 . . . . . 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 2787 . . . . 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 3143 . 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 27013 . . . . 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 6549 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun (𝐴 ↾ suc 𝑋))
67 nofun 27013 . . . . 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 6549 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → Fun (𝐵 ↾ suc 𝑋))
70 eqfunfv 6992 . . 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 2944  wral 3065  cin 3914  wss 3915  c0 4287   class class class wbr 5110  dom cdm 5638  cres 5640  Ord word 6321  Oncon0 6322  suc csuc 6324  Fun wfun 6495  cfv 6501  1oc1o 8410   No csur 27004   <s cslt 27005
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1o 8417  df-2o 8418  df-no 27007  df-slt 27008
This theorem is referenced by:  noinfbnd1lem3  27089
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