Theorem madebdaylemold 28330

Description: Lemma for madebday 28332. If the inductive hypothesis of madebday 28332 is satisfied, the converse of oldbdayim 28321 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 6914	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (bday‘𝑦) = (bday‘𝑋))
2	1	sseq1d 3944	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((bday‘𝑦) ⊆ 𝑏 ↔ (bday‘𝑋) ⊆ 𝑏))
3		eleq1 2852	. . . . . . . 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 3576	. . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) → ((bday‘𝑋) ⊆ 𝑏 → 𝑋 ∈ (M‘𝑏))))
6	5	ralimdv 3180	. . . . 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 3107	. . . 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 28184	. . . 4 ⊢ (bday‘𝑋) ∈ On
12		onelssex 6436	. . . 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 28291	. . 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 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  Oncon0 6386  ‘cfv 6564  Nocsur 28043  bdaycbday 28045  Mcmade 28254  Ocold 28255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-1o 8512  df-2o 8513  df-no 28046  df-lts 28047  df-bday 28048  df-slts 28190  df-cuts 28192  df-made 28259  df-old 28260

This theorem is used by:  madebdaylemlrcut  28331  oldbday  28333