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Theorem nolesgn2ores 27717
Description: Given 𝐴 less-than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nolesgn2ores (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

Proof of Theorem nolesgn2ores
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmres 6030 . . . 4 dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2 simp11 1204 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐴 No )
3 nodmord 27698 . . . . . . 7 (𝐴 No → Ord dom 𝐴)
42, 3syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Ord dom 𝐴)
5 ndmfv 6941 . . . . . . . . . 10 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
6 2on 8520 . . . . . . . . . . . . . . 15 2o ∈ On
76elexi 3503 . . . . . . . . . . . . . 14 2o ∈ V
87prid2 4763 . . . . . . . . . . . . 13 2o ∈ {1o, 2o}
98nosgnn0i 27704 . . . . . . . . . . . 12 ∅ ≠ 2o
10 neeq1 3003 . . . . . . . . . . . 12 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ≠ 2o ↔ ∅ ≠ 2o))
119, 10mpbiri 258 . . . . . . . . . . 11 ((𝐴𝑋) = ∅ → (𝐴𝑋) ≠ 2o)
1211neneqd 2945 . . . . . . . . . 10 ((𝐴𝑋) = ∅ → ¬ (𝐴𝑋) = 2o)
135, 12syl 17 . . . . . . . . 9 𝑋 ∈ dom 𝐴 → ¬ (𝐴𝑋) = 2o)
1413con4i 114 . . . . . . . 8 ((𝐴𝑋) = 2o𝑋 ∈ dom 𝐴)
1514adantl 481 . . . . . . 7 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) → 𝑋 ∈ dom 𝐴)
16153ad2ant2 1135 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐴)
17 ordsucss 7838 . . . . . 6 (Ord dom 𝐴 → (𝑋 ∈ dom 𝐴 → suc 𝑋 ⊆ dom 𝐴))
184, 16, 17sylc 65 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → suc 𝑋 ⊆ dom 𝐴)
19 dfss2 3969 . . . . 5 (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
2018, 19sylib 218 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
211, 20eqtrid 2789 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
22 dmres 6030 . . . 4 dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
23 simp12 1205 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐵 No )
24 nodmord 27698 . . . . . . 7 (𝐵 No → Ord dom 𝐵)
2523, 24syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Ord dom 𝐵)
26 nolesgn2o 27716 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)
27 ndmfv 6941 . . . . . . . . 9 𝑋 ∈ dom 𝐵 → (𝐵𝑋) = ∅)
28 neeq1 3003 . . . . . . . . . . 11 ((𝐵𝑋) = ∅ → ((𝐵𝑋) ≠ 2o ↔ ∅ ≠ 2o))
299, 28mpbiri 258 . . . . . . . . . 10 ((𝐵𝑋) = ∅ → (𝐵𝑋) ≠ 2o)
3029neneqd 2945 . . . . . . . . 9 ((𝐵𝑋) = ∅ → ¬ (𝐵𝑋) = 2o)
3127, 30syl 17 . . . . . . . 8 𝑋 ∈ dom 𝐵 → ¬ (𝐵𝑋) = 2o)
3231con4i 114 . . . . . . 7 ((𝐵𝑋) = 2o𝑋 ∈ dom 𝐵)
3326, 32syl 17 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐵)
34 ordsucss 7838 . . . . . 6 (Ord dom 𝐵 → (𝑋 ∈ dom 𝐵 → suc 𝑋 ⊆ dom 𝐵))
3525, 33, 34sylc 65 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → suc 𝑋 ⊆ dom 𝐵)
36 dfss2 3969 . . . . 5 (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3735, 36sylib 218 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3822, 37eqtrid 2789 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
3921, 38eqtr4d 2780 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
4021eleq2d 2827 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
41 vex 3484 . . . . . . . . 9 𝑥 ∈ V
4241elsuc 6454 . . . . . . . 8 (𝑥 ∈ suc 𝑋 ↔ (𝑥𝑋𝑥 = 𝑋))
43 simp2l 1200 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
4443fveq1d 6908 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
4544adantr 480 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
46 simpr 484 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
4746fvresd 6926 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
4846fvresd 6926 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
4945, 47, 483eqtr3d 2785 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → (𝐴𝑥) = (𝐵𝑥))
5049ex 412 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥𝑋 → (𝐴𝑥) = (𝐵𝑥)))
51 simp2r 1201 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = 2o)
5251, 26eqtr4d 2780 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
53 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
54 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
5553, 54eqeq12d 2753 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑋) = (𝐵𝑋)))
5652, 55syl5ibrcom 247 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 = 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5750, 56jaod 860 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝑥𝑋𝑥 = 𝑋) → (𝐴𝑥) = (𝐵𝑥)))
5842, 57biimtrid 242 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5958imp 406 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → (𝐴𝑥) = (𝐵𝑥))
60 simpr 484 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → 𝑥 ∈ suc 𝑋)
6160fvresd 6926 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴𝑥))
6260fvresd 6926 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐵 ↾ suc 𝑋)‘𝑥) = (𝐵𝑥))
6359, 61, 623eqtr4d 2787 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
6463ex 412 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
6540, 64sylbid 240 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥)))
6665ralrimiv 3145 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ∀𝑥 ∈ dom (𝐴 ↾ suc 𝑋)((𝐴 ↾ suc 𝑋)‘𝑥) = ((𝐵 ↾ suc 𝑋)‘𝑥))
67 nofun 27694 . . . 4 (𝐴 No → Fun 𝐴)
68 funres 6608 . . . 4 (Fun 𝐴 → Fun (𝐴 ↾ suc 𝑋))
692, 67, 683syl 18 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Fun (𝐴 ↾ suc 𝑋))
70 nofun 27694 . . . 4 (𝐵 No → Fun 𝐵)
71 funres 6608 . . . 4 (Fun 𝐵 → Fun (𝐵 ↾ suc 𝑋))
7223, 70, 713syl 18 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → Fun (𝐵 ↾ suc 𝑋))
73 eqfunfv 7056 . . 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))))
7469, 72, 73syl2anc 584 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 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))))
7539, 66, 74mpbir2and 713 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  cin 3950  wss 3951  c0 4333   class class class wbr 5143  dom cdm 5685  cres 5687  Ord word 6383  Oncon0 6384  suc csuc 6386  Fun wfun 6555  cfv 6561  1oc1o 8499  2oc2o 8500   No csur 27684   <s cslt 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688
This theorem is referenced by:  nosupbnd1lem3  27755
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