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Theorem nosupbday 27197
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 27195 . . . 4 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ 𝑆 ∈ No )
32adantr 481 . . 3 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ 𝑆 ∈ No )
4 bdayval 27140 . . 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 4533 . . . . . . . 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 2784 . . . . . . 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 5903 . . . . . 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 8473 . . . . . . . . 9 2o ∈ V
109dmsnop 6212 . . . . . . . 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 4159 . . . . . . 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 5908 . . . . . . 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 6367 . . . . . . 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 2770 . . . . . 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 2788 . . . . 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 481 . . . 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 6371 . . . . . 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 483 . . . . . . . . . . 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 27190 . . . . . . . . . . . . 13 (𝐴 βŠ† No β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2322adantr 481 . . . . . . . . . . . 12 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2423adantl 482 . . . . . . . . . . 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 3378 . . . . . . . . . . 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 583 . . . . . . . . . 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 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 3982 . . . . . . 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 27140 . . . . . . 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 27169 . . . . . . . . 9 bday : No –ontoβ†’On
35 fofn 6804 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7231 . . . . . . . 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 1466 . . . . . . 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 3982 . . . . . 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 2834 . . . . 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 7802 . . . . 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 4019 . . 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 4536 . . . . . . . 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 2784 . . . . . . 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 5903 . . . . . 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 6513 . . . . . . 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 2732 . . . . . . 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 6691 . . . . . 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 2788 . . . . 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 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†’ 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 3976 . . . . . . . . . . . . 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 27140 . . . . . . . . . . . 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 7231 . . . . . . . . . . . . . 14 (( bday Fn No ∧ 𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
5936, 58mp3an1 1448 . . . . . . . . . . . . 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 3982 . . . . . . . . . . 11 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ 𝑂)
6256, 61eqeltrrd 2834 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ dom 𝑒 ∈ 𝑂)
63 onelss 6403 . . . . . . . . . 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 3980 . . . . . . . 8 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ (𝑦 ∈ dom 𝑒 β†’ 𝑦 ∈ 𝑂))
6665adantrd 492 . . . . . . 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 3155 . . . . . 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 4064 . . . . 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 482 . . . 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 4019 . . 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 4019 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  βˆƒ!wreu 3374  βˆƒ*wrmo 3375  Vcvv 3474   βˆͺ cun 3945   βŠ† wss 3947  ifcif 4527  {csn 4627  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675   β†Ύ cres 5677   β€œ cima 5678  Ord word 6360  Oncon0 6361  suc csuc 6363  β„©cio 6490   Fn wfn 6535  β€“ontoβ†’wfo 6538  β€˜cfv 6540  β„©crio 7360  2oc2o 8456   No csur 27132   <s cslt 27133   bday cbday 27134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137
This theorem is referenced by:  noetalem1  27233
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