Theorem madebdaylemold 27866

Description: Lemma for madebday 27868. If the inductive hypothesis of madebday 27868 is satisfied, the converse of oldbdayim 27857 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 6881	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ( bday ‘𝑦) = ( bday ‘𝑋))
2	1	sseq1d 3995	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑋) ⊆ 𝑏))
3		eleq1 2823	. . . . . . . 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 3602	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
6	5	ralimdv 3155	. . . . 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 3078	. . . 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 27745	. . . 4 ⊢ ( bday ‘𝑋) ∈ On
12		onelssex 6406	. . . 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 27838	. . 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 3052  ∃wrex 3061   ⊆ wss 3931  Oncon0 6357  ‘cfv 6536   No csur 27608   bday cbday 27610   M cmade 27807   O cold 27808

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sslt 27750  df-scut 27752  df-made 27812  df-old 27813

This theorem is referenced by:  madebdaylemlrcut  27867  oldbday  27869