ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2mptx GIF version

Theorem mpt2mptx 5537
Description: Express a two-argument function as a one-argument function, or vice-versa. In this version B(x) is not assumed to be constant w.r.t x. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1 (z = ⟨x, y⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2mptx (z x A ({x} × B) ↦ 𝐶) = (x A, y B𝐷)
Distinct variable groups:   x,y,z,A   y,B,z   x,𝐶,y   z,𝐷
Allowed substitution hints:   B(x)   𝐶(z)   𝐷(x,y)

Proof of Theorem mpt2mptx
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3811 . 2 (z x A ({x} × B) ↦ 𝐶) = {⟨z, w⟩ ∣ (z x A ({x} × B) w = 𝐶)}
2 df-mpt2 5460 . . 3 (x A, y B𝐷) = {⟨⟨x, y⟩, w⟩ ∣ ((x A y B) w = 𝐷)}
3 eliunxp 4418 . . . . . . 7 (z x A ({x} × B) ↔ xy(z = ⟨x, y (x A y B)))
43anbi1i 431 . . . . . 6 ((z x A ({x} × B) w = 𝐶) ↔ (xy(z = ⟨x, y (x A y B)) w = 𝐶))
5 19.41vv 1780 . . . . . 6 (xy((z = ⟨x, y (x A y B)) w = 𝐶) ↔ (xy(z = ⟨x, y (x A y B)) w = 𝐶))
6 anass 381 . . . . . . . 8 (((z = ⟨x, y (x A y B)) w = 𝐶) ↔ (z = ⟨x, y ((x A y B) w = 𝐶)))
7 mpt2mpt.1 . . . . . . . . . . 11 (z = ⟨x, y⟩ → 𝐶 = 𝐷)
87eqeq2d 2048 . . . . . . . . . 10 (z = ⟨x, y⟩ → (w = 𝐶w = 𝐷))
98anbi2d 437 . . . . . . . . 9 (z = ⟨x, y⟩ → (((x A y B) w = 𝐶) ↔ ((x A y B) w = 𝐷)))
109pm5.32i 427 . . . . . . . 8 ((z = ⟨x, y ((x A y B) w = 𝐶)) ↔ (z = ⟨x, y ((x A y B) w = 𝐷)))
116, 10bitri 173 . . . . . . 7 (((z = ⟨x, y (x A y B)) w = 𝐶) ↔ (z = ⟨x, y ((x A y B) w = 𝐷)))
12112exbii 1494 . . . . . 6 (xy((z = ⟨x, y (x A y B)) w = 𝐶) ↔ xy(z = ⟨x, y ((x A y B) w = 𝐷)))
134, 5, 123bitr2i 197 . . . . 5 ((z x A ({x} × B) w = 𝐶) ↔ xy(z = ⟨x, y ((x A y B) w = 𝐷)))
1413opabbii 3815 . . . 4 {⟨z, w⟩ ∣ (z x A ({x} × B) w = 𝐶)} = {⟨z, w⟩ ∣ xy(z = ⟨x, y ((x A y B) w = 𝐷))}
15 dfoprab2 5494 . . . 4 {⟨⟨x, y⟩, w⟩ ∣ ((x A y B) w = 𝐷)} = {⟨z, w⟩ ∣ xy(z = ⟨x, y ((x A y B) w = 𝐷))}
1614, 15eqtr4i 2060 . . 3 {⟨z, w⟩ ∣ (z x A ({x} × B) w = 𝐶)} = {⟨⟨x, y⟩, w⟩ ∣ ((x A y B) w = 𝐷)}
172, 16eqtr4i 2060 . 2 (x A, y B𝐷) = {⟨z, w⟩ ∣ (z x A ({x} × B) w = 𝐶)}
181, 17eqtr4i 2060 1 (z x A ({x} × B) ↦ 𝐶) = (x A, y B𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  {csn 3367  cop 3370   ciun 3648  {copab 3808  cmpt 3809   × cxp 4286  {coprab 5456  cmpt2 5457
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-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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  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-iun 3650  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  mpt2mpt  5538  mpt2mptsx  5765  dmmpt2ssx  5767  fmpt2x  5768
  Copyright terms: Public domain W3C validator