Theorem madeval2 27338

Description: Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
madeval2	⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

Distinct variable group:   𝑥,𝐴,𝑎,𝑏

Proof of Theorem madeval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madeval 27337	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
2		scutcl 27293	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
3		eleq1 2822	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No ↔ 𝑥 ∈ No ))
4	3	biimpd 228	. . . . . . . 8 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No → 𝑥 ∈ No ))
5	2, 4	mpan9 508	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
6	5	rexlimivw 3152	. . . . . 6 ⊢ (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
7	6	rexlimivw 3152	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
8	7	pm4.71ri 562	. . . 4 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
9	8	abbii 2803	. . 3 ⊢ {𝑥 ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))}
10		eleq1 2822	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11		breq1 5151	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
12	10, 11	anbi12d 632	. . . . . 6 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
13	12	rexxp 5841	. . . . 5 ⊢ (∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
14		imaindm 6296	. . . . . . . 8 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s ))
15		dmscut 27302	. . . . . . . . . 10 ⊢ dom |s = <<s
16	15	ineq2i 4209	. . . . . . . . 9 ⊢ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )
17	16	imaeq2i 6056	. . . . . . . 8 ⊢ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ))
18	14, 17	eqtri 2761	. . . . . . 7 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ))
19	18	eleq2i 2826	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )))
20		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
21	20	elima 6063	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22		elin 3964	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
23	22	anbi1i 625	. . . . . . . 8 ⊢ ((𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ ((𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥))
24		anass 470	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ (𝑦 ∈ <<s ∧ 𝑦 |s 𝑥)))
25	23, 24	bitri 275	. . . . . . 7 ⊢ ((𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ (𝑦 ∈ <<s ∧ 𝑦 |s 𝑥)))
26	25	rexbii2 3091	. . . . . 6 ⊢ (∃𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥 ↔ ∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
27	19, 21, 26	3bitri 297	. . . . 5 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
28		df-br 5149	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
29	28	anbi1i 625	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
30		df-ov 7409	. . . . . . . . . 10 ⊢ (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
31	30	eqeq1i 2738	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32		scutf 27303	. . . . . . . . . . 11 ⊢ |s : <<s ⟶ No
33		ffn 6715	. . . . . . . . . . 11 ⊢ ( |s : <<s ⟶ No → |s Fn <<s )
34	32, 33	ax-mp 5	. . . . . . . . . 10 ⊢ |s Fn <<s
35		fnbrfvb 6942	. . . . . . . . . 10 ⊢ (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
36	34, 35	mpan 689	. . . . . . . . 9 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
37	31, 36	bitrid 283	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
38	37	pm5.32i 576	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
39	29, 38	bitri 275	. . . . . 6 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40	39	2rexbii 3130	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
41	13, 27, 40	3bitr4i 303	. . . 4 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))
42	41	eqabi 2870	. . 3 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = {𝑥 ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}
43		df-rab 3434	. . 3 ⊢ {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))}
44	9, 42, 43	3eqtr4i 2771	. 2 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}
45	1, 44	eqtrdi 2789	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  {crab 3433   ∩ cin 3947  𝒫 cpw 4602  ⟨cop 4634  ∪ cuni 4908   class class class wbr 5148   × cxp 5674  dom cdm 5676   “ cima 5679  Oncon0 6362   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   No csur 27133   <<s csslt 27272   |s cscut 27274   M cmade 27327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sslt 27273  df-scut 27275  df-made 27332

This theorem is referenced by:  madef  27341  elmade  27352  made0  27358  madess  27361