Theorem madebdaylemold 27809

Description: Lemma for madebday 27811. If the inductive hypothesis of madebday 27811 is satisfied, the converse of oldbdayim 27800 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 6858	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ( bday ‘𝑦) = ( bday ‘𝑋))
2	1	sseq1d 3978	. . . . . . . 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 3584	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
6	5	ralimdv 3147	. . . . 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 27688	. . . 4 ⊢ ( bday ‘𝑋) ∈ On
12		onelssex 6381	. . . 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 27781	. . 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 3914  Oncon0 6332  ‘cfv 6511   No csur 27551   bday cbday 27553   M cmade 27750   O cold 27751

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sslt 27693  df-scut 27695  df-made 27755  df-old 27756

This theorem is referenced by:  madebdaylemlrcut  27810  oldbday  27812