Theorem madess 28257

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 6054	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (M “ 𝐴) ⊆ (M “ 𝐵))
2	1	unissd 4876	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
3	2	sspwd 4569	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 ∪ (M “ 𝐴) ⊆ 𝒫 ∪ (M “ 𝐵))
4	3	adantl 487	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → 𝒫 ∪ (M “ 𝐴) ⊆ 𝒫 ∪ (M “ 𝐵))
5	4	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → 𝒫 ∪ (M “ 𝐴) ⊆ 𝒫 ∪ (M “ 𝐵))
6		ssrexv 3978	. . . . . . 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 18	. . . . . 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 3978	. . . . . . . 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 18	. . . . . . 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 3177	. . . . . 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 48	. . . . 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 486	. . . 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 4000	. . 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 28224	. . . 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 486	. . 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 28224	. . . . 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 486	. . . 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 487	. . 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 3963	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (M‘𝐴) ⊆ (M‘𝐵))
20		madef 28227	. . . . . . 7 ⊢ M:On⟶𝒫 No
21	20	fdmi 6721	. . . . . 6 ⊢ dom M = On
22	21	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23		ndmfv 6917	. . . . 5 ⊢ (¬ 𝐴 ∈ dom M → (M‘𝐴) = ∅)
24	22, 23	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (M‘𝐴) = ∅)
25		0ss 4349	. . . 4 ⊢ ∅ ⊆ (M‘𝐵)
26	24, 25	eqsstrdi 3952	. . 3 ⊢ (¬ 𝐴 ∈ On → (M‘𝐴) ⊆ (M‘𝐵))
27	26	adantr 486	. 2 ⊢ ((¬ 𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (M‘𝐴) ⊆ (M‘𝐵))
28	19, 27	pm2.61ian 824	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → (M‘𝐴) ⊆ (M‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  dom cdm 5650   “ cima 5653  Oncon0 6362  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <<s cslts 28148   |s ccuts 28150  Mcmade 28213

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 7751

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 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218

This theorem is used by:  oldssmade  28258  madebday  28291