Theorem madess 28285

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 6068	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (M “ 𝐴) ⊆ (M “ 𝐵))
2	1	unissd 4880	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
3	2	sspwd 4573	. . . . . . . . 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 3983	. . . . . . 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 3983	. . . . . . . 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 3181	. . . . . 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 4003	. . 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 28252	. . . 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 28252	. . . . 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 3968	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (M‘𝐴) ⊆ (M‘𝐵))
20		madef 28255	. . . . . . 7 ⊢ M:On⟶𝒫 No
21	20	fdmi 6746	. . . . . 6 ⊢ dom M = On
22	21	eleq2i 2856	. . . . 5 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23		ndmfv 6943	. . . . 5 ⊢ (¬ 𝐴 ∈ dom M → (M‘𝐴) = ∅)
24	22, 23	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (M‘𝐴) = ∅)
25		0ss 4353	. . . 4 ⊢ ∅ ⊆ (M‘𝐵)
26	24, 25	eqsstrdi 3957	. . 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 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  dom cdm 5656   “ cima 5659  Oncon0 6383  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <<s cslts 28176   |s ccuts 28178  Mcmade 28241

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 7778

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 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246

This theorem is used by:  oldssmade  28286  madebday  28319