ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvopab3ig GIF version

Theorem fvopab3ig 5271
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
Hypotheses
Ref Expression
fvopab3ig.1 (𝑥 = 𝐴 → (𝜑𝜓))
fvopab3ig.2 (𝑦 = 𝐵 → (𝜓𝜒))
fvopab3ig.3 (𝑥𝐶 → ∃*𝑦𝜑)
fvopab3ig.4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
Assertion
Ref Expression
fvopab3ig ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab3ig
StepHypRef Expression
1 eleq1 2114 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3ig.1 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 450 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3ig.2 . . . . . . . 8 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 445 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 4030 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
76biimpar 285 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (𝐴𝐶𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
87exp43 358 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝐴𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))))
98pm2.43a 49 . . 3 (𝐴𝐶 → (𝐵𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})))
109imp 119 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
1211fveq1i 5204 . . 3 (𝐹𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴)
13 funopab 4960 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐶𝜑))
14 fvopab3ig.3 . . . . . 6 (𝑥𝐶 → ∃*𝑦𝜑)
15 moanimv 1989 . . . . . 6 (∃*𝑦(𝑥𝐶𝜑) ↔ (𝑥𝐶 → ∃*𝑦𝜑))
1614, 15mpbir 138 . . . . 5 ∃*𝑦(𝑥𝐶𝜑)
1713, 16mpgbir 1356 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
18 funopfv 5238 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴) = 𝐵))
1917, 18ax-mp 7 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴) = 𝐵)
2012, 19syl5eq 2098 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → (𝐹𝐴) = 𝐵)
2110, 20syl6 33 1 ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wcel 1407  ∃*wmo 1915  cop 3403  {copab 3842  Fun wfun 4921  cfv 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-iota 4892  df-fun 4929  df-fv 4935
This theorem is referenced by:  fvmptg  5273  fvopab6  5289  ov6g  5663
  Copyright terms: Public domain W3C validator