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Theorem nosupbday 27069
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 27067 . . . 4 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ 𝑆 ∈ No )
32adantr 482 . . 3 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ 𝑆 ∈ No )
4 bdayval 27012 . . 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 4493 . . . . . . . 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 2785 . . . . . . 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 5862 . . . . . 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 8424 . . . . . . . . 9 2o ∈ V
109dmsnop 6169 . . . . . . . 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 4121 . . . . . . 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 5867 . . . . . . 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 6324 . . . . . . 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 2771 . . . . . 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 2789 . . . . 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 482 . . . 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 780 . . . . . 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 6328 . . . . . 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 778 . . . . . . . 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 484 . . . . . . . . . . 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 27062 . . . . . . . . . . . . 13 (𝐴 βŠ† No β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2322adantr 482 . . . . . . . . . . . 12 ((𝐴 βŠ† No ∧ 𝐴 ∈ V) β†’ βˆƒ*π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <s 𝑦)
2423adantl 483 . . . . . . . . . . 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 3354 . . . . . . . . . . 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 584 . . . . . . . . . 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 716 . . . . . . . . 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 7332 . . . . . . . . 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 3946 . . . . . . 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 27012 . . . . . . 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 781 . . . . . . 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 27041 . . . . . . . . 9 bday : No –ontoβ†’On
35 fofn 6759 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7184 . . . . . . . 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 1467 . . . . . . 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 3946 . . . . . 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 2835 . . . . 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 7754 . . . . 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 3983 . . 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 4496 . . . . . . . 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 2785 . . . . . . 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 5862 . . . . . 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 6470 . . . . . . 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 2733 . . . . . . 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 6646 . . . . . 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 2789 . . . . 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 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†’ 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 776 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑂 ∈ On)
53 ssel2 3940 . . . . . . . . . . . . 13 ((𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
5453ad4ant14 751 . . . . . . . . . . . 12 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ No )
55 bdayval 27012 . . . . . . . . . . . 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 777 . . . . . . . . . . . 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 7184 . . . . . . . . . . . . . 14 (( bday Fn No ∧ 𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
5936, 58mp3an1 1449 . . . . . . . . . . . . 13 ((𝐴 βŠ† No ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ ( bday β€œ 𝐴))
6059ad4ant14 751 . . . . . . . . . . . 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 3946 . . . . . . . . . . 11 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ ( bday β€˜π‘’) ∈ 𝑂)
6256, 61eqeltrrd 2835 . . . . . . . . . 10 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ dom 𝑒 ∈ 𝑂)
63 onelss 6360 . . . . . . . . . 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 3944 . . . . . . . 8 ((((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) ∧ 𝑒 ∈ 𝐴) β†’ (𝑦 ∈ dom 𝑒 β†’ 𝑦 ∈ 𝑂))
6665adantrd 493 . . . . . . 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 4026 . . . . 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 483 . . . 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 3983 . . 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 811 . 2 (((𝐴 βŠ† No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐴) βŠ† 𝑂)) β†’ dom 𝑆 βŠ† 𝑂)
725, 71eqsstrd 3983 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070  βˆƒ!wreu 3350  βˆƒ*wrmo 3351  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911  ifcif 4487  {csn 4587  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634   β†Ύ cres 5636   β€œ cima 5637  Ord word 6317  Oncon0 6318  suc csuc 6320  β„©cio 6447   Fn wfn 6492  β€“ontoβ†’wfo 6495  β€˜cfv 6497  β„©crio 7313  2oc2o 8407   No csur 27004   <s cslt 27005   bday cbday 27006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noetalem1  27105
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