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Theorem nolesgn2ores 31950
 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 5454 . . . 4 dom (𝐴 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐴)
2 simp11 1111 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → 𝐴 No )
3 nodmord 31931 . . . . . . 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 6256 . . . . . . . . . 10 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
6 2on 7613 . . . . . . . . . . . . . . 15 2𝑜 ∈ On
76elexi 3244 . . . . . . . . . . . . . 14 2𝑜 ∈ V
87prid2 4330 . . . . . . . . . . . . 13 2𝑜 ∈ {1𝑜, 2𝑜}
98nosgnn0i 31937 . . . . . . . . . . . 12 ∅ ≠ 2𝑜
10 neeq1 2885 . . . . . . . . . . . 12 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ≠ 2𝑜 ↔ ∅ ≠ 2𝑜))
119, 10mpbiri 248 . . . . . . . . . . 11 ((𝐴𝑋) = ∅ → (𝐴𝑋) ≠ 2𝑜)
1211neneqd 2828 . . . . . . . . . 10 ((𝐴𝑋) = ∅ → ¬ (𝐴𝑋) = 2𝑜)
135, 12syl 17 . . . . . . . . 9 𝑋 ∈ dom 𝐴 → ¬ (𝐴𝑋) = 2𝑜)
1413con4i 113 . . . . . . . 8 ((𝐴𝑋) = 2𝑜𝑋 ∈ dom 𝐴)
1514adantl 481 . . . . . . 7 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) → 𝑋 ∈ dom 𝐴)
16153ad2ant2 1103 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → 𝑋 ∈ dom 𝐴)
17 ordsucss 7060 . . . . . 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 3621 . . . . 5 (suc 𝑋 ⊆ dom 𝐴 ↔ (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
2018, 19sylib 208 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐴) = suc 𝑋)
211, 20syl5eq 2697 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = suc 𝑋)
22 dmres 5454 . . . 4 dom (𝐵 ↾ suc 𝑋) = (suc 𝑋 ∩ dom 𝐵)
23 simp12 1112 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → 𝐵 No )
24 nodmord 31931 . . . . . . 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 31949 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2𝑜)
27 ndmfv 6256 . . . . . . . . 9 𝑋 ∈ dom 𝐵 → (𝐵𝑋) = ∅)
28 neeq1 2885 . . . . . . . . . . 11 ((𝐵𝑋) = ∅ → ((𝐵𝑋) ≠ 2𝑜 ↔ ∅ ≠ 2𝑜))
299, 28mpbiri 248 . . . . . . . . . 10 ((𝐵𝑋) = ∅ → (𝐵𝑋) ≠ 2𝑜)
3029neneqd 2828 . . . . . . . . 9 ((𝐵𝑋) = ∅ → ¬ (𝐵𝑋) = 2𝑜)
3127, 30syl 17 . . . . . . . 8 𝑋 ∈ dom 𝐵 → ¬ (𝐵𝑋) = 2𝑜)
3231con4i 113 . . . . . . 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 7060 . . . . . 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 3621 . . . . 5 (suc 𝑋 ⊆ dom 𝐵 ↔ (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3735, 36sylib 208 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (suc 𝑋 ∩ dom 𝐵) = suc 𝑋)
3822, 37syl5eq 2697 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐵 ↾ suc 𝑋) = suc 𝑋)
3921, 38eqtr4d 2688 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → dom (𝐴 ↾ suc 𝑋) = dom (𝐵 ↾ suc 𝑋))
4021eleq2d 2716 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ dom (𝐴 ↾ suc 𝑋) ↔ 𝑥 ∈ suc 𝑋))
41 vex 3234 . . . . . . . . 9 𝑥 ∈ V
4241elsuc 5832 . . . . . . . 8 (𝑥 ∈ suc 𝑋 ↔ (𝑥𝑋𝑥 = 𝑋))
43 simp2l 1107 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
4443fveq1d 6231 . . . . . . . . . . . 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 476 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
4746fvresd 6246 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
4846fvresd 6246 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
4945, 47, 483eqtr3d 2693 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥𝑋) → (𝐴𝑥) = (𝐵𝑥))
5049ex 449 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝑥𝑋 → (𝐴𝑥) = (𝐵𝑥)))
51 simp2r 1108 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = 2𝑜)
5251, 26eqtr4d 2688 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐴𝑋) = (𝐵𝑋))
53 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
54 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
5553, 54eqeq12d 2666 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑋) = (𝐵𝑋)))
5652, 55syl5ibrcom 237 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 = 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5750, 56jaod 394 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → ((𝑥𝑋𝑥 = 𝑋) → (𝐴𝑥) = (𝐵𝑥)))
5842, 57syl5bi 232 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝑥 ∈ suc 𝑋 → (𝐴𝑥) = (𝐵𝑥)))
5958imp 444 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → (𝐴𝑥) = (𝐵𝑥))
60 simpr 476 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → 𝑥 ∈ suc 𝑋)
6160fvresd 6246 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) ∧ 𝑥 ∈ suc 𝑋) → ((𝐴 ↾ suc 𝑋)‘𝑥) = (𝐴𝑥))
6260fvresd 6246 . . . . . 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 2695 . . . . 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 449 . . . 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 230 . . 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 2994 . 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 31927 . . . 4 (𝐴 No → Fun 𝐴)
68 funres 5967 . . . 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 31927 . . . 4 (𝐵 No → Fun 𝐵)
71 funres 5967 . . . 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 6356 . . 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 694 . 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 977 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 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948   class class class wbr 4685  dom cdm 5143   ↾ cres 5145  Ord word 5760  Oncon0 5761  suc csuc 5763  Fun wfun 5920  ‘cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
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