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Theorem nosupbday 27593
Description: Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.)
Hypothesis
Ref Expression
nosupbday.1 𝑆 = if(βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦, ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
Assertion
Ref Expression
nosupbday (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ ( bday β€˜π‘†) βŠ† 𝑂)
Distinct variable groups:   𝐴,𝑔,𝑒,𝑣,π‘₯,𝑦   𝑒,𝑂,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦,𝑣,𝑒,𝑔)   𝑂(π‘₯,𝑣,𝑔)
Proof of Theorem nosupbday
StepHypRef Expression
1 nosupbday.1 . . . . 5 𝑆 = if(βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦, ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
21nosupno 27591 . . . 4 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ 𝑆 ∈ No )
32adantr 480 . . 3 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ 𝑆 ∈ No )
4 bdayval 27536 . . 3 (𝑆 ∈ No β†’ ( bday β€˜π‘†) = dom 𝑆)
53, 4syl 17 . 2 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ ( bday β€˜π‘†) = dom 𝑆)
6 iftrue 4529 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ if(βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦, ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯)))) = ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}))
71, 6eqtrid 2778 . . . . . . 7 (βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ 𝑆 = ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}))
87dmeqd 5899 . . . . . 6 (βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ dom 𝑆 = dom ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}))
9 2oex 8478 . . . . . . . . 9 2o ∈ V
109dmsnop 6209 . . . . . . . 8 dom {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩} = {dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)}
1110uneq2i 4155 . . . . . . 7 (dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ dom {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}) = (dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)})
12 dmun 5904 . . . . . . 7 dom ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}) = (dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ dom {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩})
13 df-suc 6364 . . . . . . 7 suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) = (dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)})
1411, 12, 133eqtr4i 2764 . . . . . 6 dom ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}) = suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
158, 14eqtrdi 2782 . . . . 5 (βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ dom 𝑆 = suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦))
1615adantr 480 . . . 4 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ dom 𝑆 = suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦))
17 simprrl 778 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ 𝑂 ∈ On)
18 eloni 6368 . . . . . 6 (𝑂 ∈ On β†’ Ord 𝑂)
1917, 18syl 17 . . . . 5 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ Ord 𝑂)
20 simprll 776 . . . . . . . 8 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ 𝐴 βŠ† No )
21 simpl 482 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ (𝐴 βŠ† No ∧ 𝐴 ∈ V)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
22 nomaxmo 27586 . . . . . . . . . . . . 13 (𝐴 βŠ† No β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2322adantr 480 . . . . . . . . . . . 12 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2423adantl 481 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ (𝐴 βŠ† No ∧ 𝐴 ∈ V)) β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
25 reu5 3372 . . . . . . . . . . 11 (βˆƒ!π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ↔ (βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦))
2621, 24, 25sylanbrc 582 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ (𝐴 βŠ† No ∧ 𝐴 ∈ V)) β†’ βˆƒ!π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2726adantrr 714 . . . . . . . . 9 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ βˆƒ!π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
28 riotacl 7379 . . . . . . . . 9 (βˆƒ!π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ 𝐴)
2927, 28syl 17 . . . . . . . 8 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ 𝐴)
3020, 29sseldd 3978 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ No )
31 bdayval 27536 . . . . . . 7 ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ No β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)) = dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦))
3230, 31syl 17 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)) = dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦))
33 simprrr 779 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ ( bday β€œ 𝐴) βŠ† 𝑂)
34 bdayfo 27565 . . . . . . . . 9 bday : No –ontoβ†’On
35 fofn 6801 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7230 . . . . . . . 8 (( bday Fn No ∧ 𝐴 βŠ† No ∧ (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ 𝐴) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)) ∈ ( bday β€œ 𝐴))
3836, 20, 29, 37mp3an2i 1462 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)) ∈ ( bday β€œ 𝐴))
3933, 38sseldd 3978 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)) ∈ 𝑂)
4032, 39eqeltrrd 2828 . . . . 5 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ 𝑂)
41 ordsucss 7803 . . . . 5 (Ord 𝑂 β†’ (dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) ∈ 𝑂 β†’ suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βŠ† 𝑂))
4219, 40, 41sylc 65 . . . 4 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ suc dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βŠ† 𝑂)
4316, 42eqsstrd 4015 . . 3 ((βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ dom 𝑆 βŠ† 𝑂)
44 iffalse 4532 . . . . . . . 8 (Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ if(βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦, ((β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯)))) = (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
451, 44eqtrid 2778 . . . . . . 7 (Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ 𝑆 = (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
4645dmeqd 5899 . . . . . 6 (Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ dom 𝑆 = dom (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
47 iotaex 6510 . . . . . . 7 (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯)) ∈ V
48 eqid 2726 . . . . . . 7 (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))) = (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯)))
4947, 48dmmpti 6688 . . . . . 6 dom (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐴 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))) = {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))}
5046, 49eqtrdi 2782 . . . . 5 (Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 β†’ dom 𝑆 = {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))})
5150adantr 480 . . . 4 ((Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ dom 𝑆 = {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))})
52 simplrl 774 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑂 ∈ On)
53 ssel2 3972 . . . . . . . . . . . . 13 ((𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
5453ad4ant14 749 . . . . . . . . . . . 12 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
55 bdayval 27536 . . . . . . . . . . . 12 (𝑒 ∈ No β†’ ( bday β€˜π‘’) = dom 𝑒)
5654, 55syl 17 . . . . . . . . . . 11 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) = dom 𝑒)
57 simplrr 775 . . . . . . . . . . . 12 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€œ 𝐴) βŠ† 𝑂)
58 fnfvima 7230 . . . . . . . . . . . . . 14 (( bday Fn No ∧ 𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
5936, 58mp3an1 1444 . . . . . . . . . . . . 13 ((𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
6059ad4ant14 749 . . . . . . . . . . . 12 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
6157, 60sseldd 3978 . . . . . . . . . . 11 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ 𝑂)
6256, 61eqeltrrd 2828 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ dom 𝑒 ∈ 𝑂)
63 onelss 6400 . . . . . . . . . 10 (𝑂 ∈ On β†’ (dom 𝑒 ∈ 𝑂 β†’ dom 𝑒 βŠ† 𝑂))
6452, 62, 63sylc 65 . . . . . . . . 9 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ dom 𝑒 βŠ† 𝑂)
6564sseld 3976 . . . . . . . 8 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ (𝑦 ∈ dom 𝑒 β†’ 𝑦 ∈ 𝑂))
6665adantrd 491 . . . . . . 7 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ((𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦))) β†’ 𝑦 ∈ 𝑂))
6766rexlimdva 3149 . . . . . 6 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ (βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦))) β†’ 𝑦 ∈ 𝑂))
6867abssdv 4060 . . . . 5 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} βŠ† 𝑂)
6968adantl 481 . . . 4 ((Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐴 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐴 (Β¬ 𝑣 <s 𝑒 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} βŠ† 𝑂)
7051, 69eqsstrd 4015 . . 3 ((Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦 ∧ ((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂))) β†’ dom 𝑆 βŠ† 𝑂)
7143, 70pm2.61ian 809 . 2 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ dom 𝑆 βŠ† 𝑂)
725, 71eqsstrd 4015 1 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ ( bday β€˜π‘†) βŠ† 𝑂)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  βˆƒwrex 3064  βˆƒ!wreu 3368  βˆƒ*wrmo 3369  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  ifcif 4523  {csn 4623  βŸ¨cop 4629   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669   β†Ύ cres 5671   β€œ cima 5672  Ord word 6357  Oncon0 6358  suc csuc 6360  β„©cio 6487   Fn wfn 6532  β€“ontoβ†’wfo 6535  β€˜cfv 6537  β„©crio 7360  2oc2o 8461   No csur 27528   <s cslt 27529   bday cbday 27530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533
This theorem is referenced by:  noetalem1  27629
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