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Theorem nosupbday 27656
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 27654 . . . 4 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ 𝑆 ∈ No )
32adantr 479 . . 3 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ 𝑆 ∈ No )
4 bdayval 27599 . . 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 4530 . . . . . . . 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 2777 . . . . . . 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 5902 . . . . . 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 8496 . . . . . . . . 9 2o ∈ V
109dmsnop 6215 . . . . . . . 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 4153 . . . . . . 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 5907 . . . . . . 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 6370 . . . . . . 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 2763 . . . . . 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 2781 . . . . 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 479 . . . 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 779 . . . . . 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 6374 . . . . . 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 777 . . . . . . . 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 481 . . . . . . . . . . 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 27649 . . . . . . . . . . . . 13 (𝐴 βŠ† No β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2322adantr 479 . . . . . . . . . . . 12 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2423adantl 480 . . . . . . . . . . 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 3366 . . . . . . . . . . 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 581 . . . . . . . . . 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 715 . . . . . . . . 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 7390 . . . . . . . . 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 3973 . . . . . . 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 27599 . . . . . . 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 780 . . . . . . 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 27628 . . . . . . . . 9 bday : No –ontoβ†’On
35 fofn 6808 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7241 . . . . . . . 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 3973 . . . . . 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 2826 . . . . 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 7819 . . . . 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 4011 . . 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 4533 . . . . . . . 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 2777 . . . . . . 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 5902 . . . . . 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 6516 . . . . . . 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 2725 . . . . . . 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 6694 . . . . . 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 2781 . . . . 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 479 . . . 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 775 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑂 ∈ On)
53 ssel2 3967 . . . . . . . . . . . . 13 ((𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
5453ad4ant14 750 . . . . . . . . . . . 12 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
55 bdayval 27599 . . . . . . . . . . . 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 776 . . . . . . . . . . . 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 7241 . . . . . . . . . . . . . 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 750 . . . . . . . . . . . 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 3973 . . . . . . . . . . 11 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ 𝑂)
6256, 61eqeltrrd 2826 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ dom 𝑒 ∈ 𝑂)
63 onelss 6406 . . . . . . . . . 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 3971 . . . . . . . 8 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ (𝑦 ∈ dom 𝑒 β†’ 𝑦 ∈ 𝑂))
6665adantrd 490 . . . . . . 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 3145 . . . . . 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 4057 . . . . 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 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†’ {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 4011 . . 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 810 . 2 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ dom 𝑆 βŠ† 𝑂)
725, 71eqsstrd 4011 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆ€wral 3051  βˆƒwrex 3060  βˆƒ!wreu 3362  βˆƒ*wrmo 3363  Vcvv 3463   βˆͺ cun 3937   βŠ† wss 3939  ifcif 4524  {csn 4624  βŸ¨cop 4630   class class class wbr 5143   ↦ cmpt 5226  dom cdm 5672   β†Ύ cres 5674   β€œ cima 5675  Ord word 6363  Oncon0 6364  suc csuc 6366  β„©cio 6493   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  β„©crio 7371  2oc2o 8479   No csur 27591   <s cslt 27592   bday cbday 27593
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-1o 8485  df-2o 8486  df-no 27594  df-slt 27595  df-bday 27596
This theorem is referenced by:  noetalem1  27692
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