Theorem mpt2mptsx 5765

Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
mpt2mptsx	⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)

Distinct variable groups:   x,y,z,A   y,B,z   z,𝐶

Allowed substitution hints:   B(x)   𝐶(x,y)

Proof of Theorem mpt2mptsx

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2554	. . . . . 6 ⊢ u ∈ V
2		vex 2554	. . . . . 6 ⊢ v ∈ V
3	1, 2	op1std 5717	. . . . 5 ⊢ (z = ⟨u, v⟩ → (1st ‘z) = u)
4	3	csbeq1d 2852	. . . 4 ⊢ (z = ⟨u, v⟩ → ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 = ⦋u / x⦌⦋(2nd ‘z) / y⦌𝐶)
5	1, 2	op2ndd 5718	. . . . . 6 ⊢ (z = ⟨u, v⟩ → (2nd ‘z) = v)
6	5	csbeq1d 2852	. . . . 5 ⊢ (z = ⟨u, v⟩ → ⦋(2nd ‘z) / y⦌𝐶 = ⦋v / y⦌𝐶)
7	6	csbeq2dv 2869	. . . 4 ⊢ (z = ⟨u, v⟩ → ⦋u / x⦌⦋(2nd ‘z) / y⦌𝐶 = ⦋u / x⦌⦋v / y⦌𝐶)
8	4, 7	eqtrd 2069	. . 3 ⊢ (z = ⟨u, v⟩ → ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 = ⦋u / x⦌⦋v / y⦌𝐶)
9	8	mpt2mptx 5537	. 2 ⊢ (z ∈ ∪ u ∈ A ({u} × ⦋u / x⦌B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶) = (u ∈ A, v ∈ ⦋u / x⦌B ↦ ⦋u / x⦌⦋v / y⦌𝐶)
10		nfcv 2175	. . . 4 ⊢ Ⅎu({x} × B)
11		nfcv 2175	. . . . 5 ⊢ Ⅎx{u}
12		nfcsb1v 2876	. . . . 5 ⊢ Ⅎx⦋u / x⦌B
13	11, 12	nfxp 4314	. . . 4 ⊢ Ⅎx({u} × ⦋u / x⦌B)
14		sneq 3378	. . . . 5 ⊢ (x = u → {x} = {u})
15		csbeq1a 2854	. . . . 5 ⊢ (x = u → B = ⦋u / x⦌B)
16	14, 15	xpeq12d 4313	. . . 4 ⊢ (x = u → ({x} × B) = ({u} × ⦋u / x⦌B))
17	10, 13, 16	cbviun 3685	. . 3 ⊢ ∪ x ∈ A ({x} × B) = ∪ u ∈ A ({u} × ⦋u / x⦌B)
18		mpteq1 3832	. . 3 ⊢ (∪ x ∈ A ({x} × B) = ∪ u ∈ A ({u} × ⦋u / x⦌B) → (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶) = (z ∈ ∪ u ∈ A ({u} × ⦋u / x⦌B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶))
19	17, 18	ax-mp 7	. 2 ⊢ (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶) = (z ∈ ∪ u ∈ A ({u} × ⦋u / x⦌B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
20		nfcv 2175	. . 3 ⊢ ℲuB
21		nfcv 2175	. . 3 ⊢ Ⅎu𝐶
22		nfcv 2175	. . 3 ⊢ Ⅎv𝐶
23		nfcsb1v 2876	. . 3 ⊢ Ⅎx⦋u / x⦌⦋v / y⦌𝐶
24		nfcv 2175	. . . 4 ⊢ Ⅎyu
25		nfcsb1v 2876	. . . 4 ⊢ Ⅎy⦋v / y⦌𝐶
26	24, 25	nfcsb 2878	. . 3 ⊢ Ⅎy⦋u / x⦌⦋v / y⦌𝐶
27		csbeq1a 2854	. . . 4 ⊢ (y = v → 𝐶 = ⦋v / y⦌𝐶)
28		csbeq1a 2854	. . . 4 ⊢ (x = u → ⦋v / y⦌𝐶 = ⦋u / x⦌⦋v / y⦌𝐶)
29	27, 28	sylan9eqr 2091	. . 3 ⊢ ((x = u ∧ y = v) → 𝐶 = ⦋u / x⦌⦋v / y⦌𝐶)
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpt2x 5524	. 2 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (u ∈ A, v ∈ ⦋u / x⦌B ↦ ⦋u / x⦌⦋v / y⦌𝐶)
31	9, 19, 30	3eqtr4ri 2068	1 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)

Syntax hints:   = wceq 1242  ⦋csb 2846  {csn 3367  ⟨cop 3370  ∪ ciun 3648   ↦ cmpt 3809   × cxp 4286  ‘cfv 4845   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by:  mpt2mpts  5766  mpt2fvex  5771