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Theorem nolesgn2ores 33292
 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 5840 . . . 4 dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2 simp11 1200 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐴 No )
3 nodmord 33273 . . . . . . 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 6675 . . . . . . . . . 10 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
6 2on 8094 . . . . . . . . . . . . . . 15 2o ∈ On
76elexi 3460 . . . . . . . . . . . . . 14 2o ∈ V
87prid2 4659 . . . . . . . . . . . . 13 2o ∈ {1o, 2o}
98nosgnn0i 33279 . . . . . . . . . . . 12 ∅ ≠ 2o
10 neeq1 3049 . . . . . . . . . . . 12 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ≠ 2o ↔ ∅ ≠ 2o))
119, 10mpbiri 261 . . . . . . . . . . 11 ((𝐴𝑋) = ∅ → (𝐴𝑋) ≠ 2o)
1211neneqd 2992 . . . . . . . . . 10 ((𝐴𝑋) = ∅ → ¬ (𝐴𝑋) = 2o)
135, 12syl 17 . . . . . . . . 9 𝑋 ∈ dom 𝐴 → ¬ (𝐴𝑋) = 2o)
1413con4i 114 . . . . . . . 8 ((𝐴𝑋) = 2o𝑋 ∈ dom 𝐴)
1514adantl 485 . . . . . . 7 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) → 𝑋 ∈ dom 𝐴)
16153ad2ant2 1131 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐴)
17 ordsucss 7513 . . . . . 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 df-ss 3898 . . . . 5 (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
2018, 19sylib 221 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
211, 20syl5eq 2845 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
22 dmres 5840 . . . 4 dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
23 simp12 1201 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → 𝐵 No )
24 nodmord 33273 . . . . . . 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 33291 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)
27 ndmfv 6675 . . . . . . . . 9 𝑋 ∈ dom 𝐵 → (𝐵𝑋) = ∅)
28 neeq1 3049 . . . . . . . . . . 11 ((𝐵𝑋) = ∅ → ((𝐵𝑋) ≠ 2o ↔ ∅ ≠ 2o))
299, 28mpbiri 261 . . . . . . . . . 10 ((𝐵𝑋) = ∅ → (𝐵𝑋) ≠ 2o)
3029neneqd 2992 . . . . . . . . 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 7513 . . . . . 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 df-ss 3898 . . . . 5 (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3735, 36sylib 221 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3822, 37syl5eq 2845 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
3921, 38eqtr4d 2836 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
4021eleq2d 2875 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
41 vex 3444 . . . . . . . . 9 𝑥 ∈ V
4241elsuc 6228 . . . . . . . 8 (𝑥 ∈ suc 𝑋 ↔ (𝑥𝑋𝑥 = 𝑋))
43 simp2l 1196 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
4443fveq1d 6647 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
4544adantr 484 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
46 simpr 488 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
4746fvresd 6665 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
4846fvresd 6665 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
4945, 47, 483eqtr3d 2841 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → (𝐴𝑥) = (𝐵𝑥))
5049ex 416 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥𝑋 → (𝐴𝑥) = (𝐵𝑥)))
51 simp2r 1197 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = 2o)
5251, 26eqtr4d 2836 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
53 fveq2 6645 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
54 fveq2 6645 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
5553, 54eqeq12d 2814 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑋) = (𝐵𝑋)))
5652, 55syl5ibrcom 250 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 = 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5750, 56jaod 856 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → ((𝑥𝑋𝑥 = 𝑋) → (𝐴𝑥) = (𝐵𝑥)))
5842, 57syl5bi 245 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5958imp 410 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → (𝐴𝑥) = (𝐵𝑥))
60 simpr 488 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → 𝑥 ∈ suc 𝑋)
6160fvresd 6665 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴𝑥))
6260fvresd 6665 . . . . . 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 2843 . . . . 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 416 . . . 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 243 . . 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 3148 . 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 33269 . . . 4 (𝐴 No → Fun 𝐴)
68 funres 6366 . . . 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 33269 . . . 4 (𝐵 No → Fun 𝐵)
71 funres 6366 . . . 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 6784 . . 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 587 . 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 712 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243   class class class wbr 5030  dom cdm 5519   ↾ cres 5521  Ord word 6158  Oncon0 6159  suc csuc 6161  Fun wfun 6318  ‘cfv 6324  1oc1o 8078  2oc2o 8079   No csur 33260
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