Theorem madess 27719

Description: If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)

Assertion

Ref	Expression
madess	⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Proof of Theorem madess

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imass2 6091	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4909	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	sspwd 4607	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
5	4	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
6		ssrexv 4043	. . . . . . 7 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
8		ssrexv 4043	. . . . . . . 8 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
9	5, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
10	9	reximdv 3162	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
11	7, 10	syld 47	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
12	11	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) ∧ 𝑥 ∈ No ) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
13	12	ss2rabdv 4065	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} ⊆ {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
14		madeval2 27696	. . . 4 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
15	14	adantr 480	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
16		madeval2 27696	. . . . 5 ⊢ (𝐵 ∈ On → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
17	16	adantr 480	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
18	17	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
19	13, 15, 18	3sstr4d 4021	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
20		madef 27699	. . . . . . 7 ⊢ M :On⟶𝒫 No
21	20	fdmi 6719	. . . . . 6 ⊢ dom M = On
22	21	eleq2i 2817	. . . . 5 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23		ndmfv 6916	. . . . 5 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
24	22, 23	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
25		0ss 4388	. . . 4 ⊢ ∅ ⊆ ( M ‘𝐵)
26	24, 25	eqsstrdi 4028	. . 3 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) ⊆ ( M ‘𝐵))
27	26	adantr 480	. 2 ⊢ ((¬ 𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
28	19, 27	pm2.61ian 809	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∃wrex 3062  {crab 3424   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  ∪ cuni 4899   class class class wbr 5138  dom cdm 5666   “ cima 5669  Oncon0 6354  ‘cfv 6533  (class class class)co 7401   No csur 27489   <<s csslt 27629   |s cscut 27631   M cmade 27685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-2o 8462  df-no 27492  df-slt 27493  df-bday 27494  df-sslt 27630  df-scut 27632  df-made 27690

This theorem is referenced by:  oldssmade  27720  madebday  27742