Theorem madess 27824

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 6057	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4870	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	sspwd 4564	. . . . . . . . 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 4000	. . . . . . 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 4000	. . . . . . . 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 3148	. . . . . 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 4024	. . 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 27797	. . . 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 27797	. . . . 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 3986	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
20		madef 27800	. . . . . . 7 ⊢ M :On⟶𝒫 No
21	20	fdmi 6669	. . . . . 6 ⊢ dom M = On
22	21	eleq2i 2825	. . . . 5 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23		ndmfv 6862	. . . . 5 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
24	22, 23	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
25		0ss 4349	. . . 4 ⊢ ∅ ⊆ ( M ‘𝐵)
26	24, 25	eqsstrdi 3975	. . 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 811	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  {crab 3396   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4551  ∪ cuni 4860   class class class wbr 5095  dom cdm 5621   “ cima 5624  Oncon0 6313  ‘cfv 6488  (class class class)co 7354   No csur 27581   <<s csslt 27723   |s cscut 27725   M cmade 27786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-1o 8393  df-2o 8394  df-no 27584  df-slt 27585  df-bday 27586  df-sslt 27724  df-scut 27726  df-made 27791

This theorem is referenced by:  oldssmade  27825  madebday  27848