Theorem madebdaylemold 27890

Description: Lemma for madebday 27892. If the inductive hypothesis of madebday 27892 is satisfied, the converse of oldbdayim 27881 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 6840	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ( bday ‘𝑦) = ( bday ‘𝑋))
2	1	sseq1d 3953	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑋) ⊆ 𝑏))
3		eleq1 2824	. . . . . . . 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 3560	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))))
6	5	ralimdv 3151	. . . . 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 1131	. 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		bdayon 27744	. . . 4 ⊢ ( bday ‘𝑋) ∈ On
12		onelssex 6372	. . . 4 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏))
13	11, 12	mpan 691	. . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏))
14	13	3ad2ant1 1134	. 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 27851	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
16	15	3ad2ant1 1134	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  Oncon0 6323  ‘cfv 6498   No csur 27603   bday cbday 27605   M cmade 27814   O cold 27815

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820

This theorem is referenced by:  madebdaylemlrcut  27891  oldbday  27893