Theorem elmade 27865

Description: Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
elmade	⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))

Distinct variable groups:   𝐴,𝑙,𝑟   𝑋,𝑙,𝑟

Proof of Theorem elmade

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madef 27844	. . . . 5 ⊢ M :On⟶𝒫 No
2	1	ffvelcdmi 7037	. . . 4 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ∈ 𝒫 No )
3	2	elpwid 4565	. . 3 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ⊆ No )
4	3	sseld 3934	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → 𝑋 ∈ No ))
5		cutscl 27790	. . . . . 6 ⊢ (𝑙 <<s 𝑟 → (𝑙 |s 𝑟) ∈ No )
6		eleq1 2825	. . . . . . 7 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No ↔ 𝑋 ∈ No ))
7	6	biimpd 229	. . . . . 6 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No → 𝑋 ∈ No ))
8	5, 7	mpan9 506	. . . . 5 ⊢ ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
9	8	rexlimivw 3135	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
10	9	rexlimivw 3135	. . 3 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
11	10	a1i 11	. 2 ⊢ (𝐴 ∈ On → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No ))
12		madeval2 27841	. . . . 5 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
13	12	eleq2d 2823	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)}))
14		eqeq2 2749	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s 𝑟) = 𝑋))
15	14	anbi2d 631	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
16	15	2rexbidv 3203	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
17	16	elrab3 3649	. . . 4 ⊢ (𝑋 ∈ No → (𝑋 ∈ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
18	13, 17	sylan9bb 509	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
19	18	ex 412	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ No → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋))))
20	4, 11, 19	pm5.21ndd 379	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100   “ cima 5635  Oncon0 6325  ‘cfv 6500  (class class class)co 7368   No csur 27619   <<s cslts 27765   |s ccuts 27767   M cmade 27830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768  df-made 27835

This theorem is referenced by:  elmade2  27866