Theorem madebdaylemold 27830

Description: Lemma for madebday 27832. If the inductive hypothesis of madebday 27832 is satisfied, the converse of oldbdayim 27821 holds. (Contributed by Scott Fenton, 19-Aug-2024.)

Assertion

Ref	Expression
madebdaylemold	⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))

Distinct variable groups:   𝐴,𝑏   𝑦,𝑏,𝑋

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem madebdaylemold

Step	Hyp	Ref	Expression
1		fveq2 6826	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ( bday ‘𝑦) = ( bday ‘𝑋))
2	1	sseq1d 3969	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑋) ⊆ 𝑏))
3		eleq1 2816	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑋 ∈ ( M ‘𝑏)))
4	2, 3	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
5	4	rspcv 3575	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
6	5	ralimdv 3143	. . . . 5 ⊢ (𝑋 ∈ No → (∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → ∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
7	6	impcom 407	. . . 4 ⊢ ((∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)))
8		rexim 3070	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
9	7, 8	syl 17	. . 3 ⊢ ((∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
10	9	3adant1 1130	. 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
11		bdayelon 27704	. . . 4 ⊢ ( bday ‘𝑋) ∈ On
12		onelssex 6360	. . . 4 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏))
13	11, 12	mpan 690	. . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏))
14	13	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏))
15		elold 27801	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
16	15	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
17	10, 14, 16	3imtr4d 294	1 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905  Oncon0 6311  ‘cfv 6486   No csur 27567   bday cbday 27569   M cmade 27770   O cold 27771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27570  df-slt 27571  df-bday 27572  df-sslt 27710  df-scut 27712  df-made 27775  df-old 27776

This theorem is referenced by:  madebdaylemlrcut  27831  oldbday  27833