Theorem madeval2 28224

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 28223	. 2 ⊢ (𝐴 ∈ On → (M‘𝐴) = ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))))
2		cutscl 28173	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No)
3		eleq1 2848	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No ↔ 𝑥 ∈ No))
4	3	biimpd 232	. . . . . . . 8 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No → 𝑥 ∈ No))
5	2, 4	mpan9 516	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
6	5	rexlimivw 3159	. . . . . 6 ⊢ (∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
7	6	rexlimivw 3159	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ (M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
8	7	pm4.71ri 570	. . . 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 2827	. . 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 2848	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11		breq1 5105	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
12	10, 11	anbi12d 644	. . . . . 6 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
13	12	rexxp 5818	. . . . 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 6302	. . . . . . . 8 ⊢ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s ))
15		dmcuts 28182	. . . . . . . . . 10 ⊢ dom |s = <<s
16	15	ineq2i 4114	. . . . . . . . 9 ⊢ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s ) = ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )
17	16	imaeq2i 6049	. . . . . . . 8 ⊢ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ))
18	14, 17	eqtri 2783	. . . . . . 7 ⊢ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ))
19	18	eleq2i 2852	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )))
20		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
21	20	elima 6060	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22		elin 4093	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
23	22	anbi1i 636	. . . . . . . 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 474	. . . . . . . 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 278	. . . . . . 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 3105	. . . . . 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 300	. . . . 5 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
28		df-br 5103	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
29	28	anbi1i 636	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
30		df-ov 7423	. . . . . . . . . 10 ⊢ (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
31	30	eqeq1i 2765	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32		cutsf 28183	. . . . . . . . . . 11 ⊢ |s : <<s ⟶No
33		ffn 6709	. . . . . . . . . . 11 ⊢ ( |s : <<s ⟶No → |s Fn <<s )
34	32, 33	ax-mp 5	. . . . . . . . . 10 ⊢ |s Fn <<s
35		fnbrfvb 6935	. . . . . . . . . 10 ⊢ (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
36	34, 35	mpan 703	. . . . . . . . 9 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
37	31, 36	bitrid 286	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
38	37	pm5.32i 585	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
39	29, 38	bitri 278	. . . . . 6 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40	39	2rexbii 3138	. . . . 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 306	. . . 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 2895	. . 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 3413	. . 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 2793	. 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 2811	1 ⊢ (𝐴 ∈ On → (M‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ (M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  {crab 3412   ∩ cin 4089  𝒫 cpw 4556  ⟨cop 4834  ∪ cuni 4866   class class class wbr 5102   × cxp 5648  dom cdm 5650   “ cima 5653  Oncon0 6362   Fn wfn 6533  ⟶wf 6534  ‘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:  madef  28227  elmade  28248  made0  28254  madess  28257