Theorem madebdaylemold 28290

Description: Lemma for madebday 28292. If the inductive hypothesis of madebday 28292 is satisfied, the converse of oldbdayim 28281 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 6886	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (bday‘𝑦) = (bday‘𝑋))
2	1	sseq1d 3939	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((bday‘𝑦) ⊆ 𝑏 ↔ (bday‘𝑋) ⊆ 𝑏))
3		eleq1 2848	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦 ∈ (M‘𝑏) ↔ 𝑋 ∈ (M‘𝑏)))
4	2, 3	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ↔ ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏))))
5	4	rspcv 3572	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) → ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏))))
6	5	ralimdv 3176	. . . . 5 ⊢ (𝑋 ∈ No → (∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) → ∀𝑏 ∈ 𝐴 ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏))))
7	6	impcom 413	. . . 4 ⊢ ((∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → ∀𝑏 ∈ 𝐴 ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏)))
8		rexim 3103	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏)) → (∃𝑏 ∈ 𝐴 (bday‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))
9	7, 8	syl 18	. . 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 1148	. 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 28144	. . . 4 ⊢ (bday‘𝑋) ∈ On
12		onelssex 6412	. . . 4 ⊢ (((bday‘𝑋) ∈ On ∧ 𝐴 ∈ On) → ((bday‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 (bday‘𝑋) ⊆ 𝑏))
13	11, 12	mpan 703	. . 3 ⊢ (𝐴 ∈ On → ((bday‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 (bday‘𝑋) ⊆ 𝑏))
14	13	3ad2ant1 1151	. 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 28251	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ (O‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))
16	15	3ad2ant1 1151	. 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 297	1 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → ((bday‘𝑋) ∈ 𝐴 → 𝑋 ∈ (O‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  Oncon0 6362  ‘cfv 6539  Nocsur 28003  bdaycbday 28005  Mcmade 28214  Ocold 28215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219  df-old 28220

This theorem is used by:  madebdaylemlrcut  28291  oldbday  28293