ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvopab3g Structured version   GIF version

Theorem fvopab3g 5188
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
fvopab3g.2 (x = A → (φψ))
fvopab3g.3 (y = B → (ψχ))
fvopab3g.4 (x 𝐶∃!yφ)
fvopab3g.5 𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}
Assertion
Ref Expression
fvopab3g ((A 𝐶 B 𝐷) → ((𝐹A) = Bχ))
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 fvopab3g
StepHypRef Expression
1 eleq1 2097 . . . 4 (x = A → (x 𝐶A 𝐶))
2 fvopab3g.2 . . . 4 (x = A → (φψ))
31, 2anbi12d 442 . . 3 (x = A → ((x 𝐶 φ) ↔ (A 𝐶 ψ)))
4 fvopab3g.3 . . . 4 (y = B → (ψχ))
54anbi2d 437 . . 3 (y = B → ((A 𝐶 ψ) ↔ (A 𝐶 χ)))
63, 5opelopabg 3996 . 2 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)} ↔ (A 𝐶 χ)))
7 fvopab3g.4 . . . . . 6 (x 𝐶∃!yφ)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}
97, 8fnopab 4966 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 5158 . . . . 5 ((𝐹 Fn 𝐶 A 𝐶) → ((𝐹A) = B ↔ ⟨A, B 𝐹))
119, 10mpan 400 . . . 4 (A 𝐶 → ((𝐹A) = B ↔ ⟨A, B 𝐹))
128eleq2i 2101 . . . 4 (⟨A, B 𝐹 ↔ ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)})
1311, 12syl6bb 185 . . 3 (A 𝐶 → ((𝐹A) = B ↔ ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)}))
1413adantr 261 . 2 ((A 𝐶 B 𝐷) → ((𝐹A) = B ↔ ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)}))
15 ibar 285 . . 3 (A 𝐶 → (χ ↔ (A 𝐶 χ)))
1615adantr 261 . 2 ((A 𝐶 B 𝐷) → (χ ↔ (A 𝐶 χ)))
176, 14, 163bitr4d 209 1 ((A 𝐶 B 𝐷) → ((𝐹A) = Bχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  cop 3370  {copab 3808   Fn wfn 4840  cfv 4845
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-bnd 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-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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  recmulnqg  6375
  Copyright terms: Public domain W3C validator