ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvf1olem GIF version

Theorem cnvf1olem 6005
Description: Lemma for cnvf1o 6006. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1olem ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))

Proof of Theorem cnvf1olem
StepHypRef Expression
1 simprr 500 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {𝐵})
2 1st2nd 5967 . . . . . . . 8 ((Rel 𝐴𝐵𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
32adantrr 464 . . . . . . 7 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
43sneqd 3465 . . . . . 6 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
54cnveqd 4627 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
65unieqd 3672 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
7 1stexg 5954 . . . . . 6 (𝐵𝐴 → (1st𝐵) ∈ V)
8 2ndexg 5955 . . . . . 6 (𝐵𝐴 → (2nd𝐵) ∈ V)
9 opswapg 4932 . . . . . 6 (((1st𝐵) ∈ V ∧ (2nd𝐵) ∈ V) → {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
107, 8, 9syl2anc 404 . . . . 5 (𝐵𝐴 {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
1110ad2antrl 475 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
121, 6, 113eqtrd 2125 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = ⟨(2nd𝐵), (1st𝐵)⟩)
13 simprl 499 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵𝐴)
143, 13eqeltrrd 2166 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
15 opelcnvg 4631 . . . . . 6 (((2nd𝐵) ∈ V ∧ (1st𝐵) ∈ V) → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
168, 7, 15syl2anc 404 . . . . 5 (𝐵𝐴 → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
1716ad2antrl 475 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
1814, 17mpbird 166 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴)
1912, 18eqeltrd 2165 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶𝐴)
20 opswapg 4932 . . . . . 6 (((2nd𝐵) ∈ V ∧ (1st𝐵) ∈ V) → {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩)
218, 7, 20syl2anc 404 . . . . 5 (𝐵𝐴 {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩)
2221eqcomd 2094 . . . 4 (𝐵𝐴 → ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩})
2322ad2antrl 475 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩})
2412sneqd 3465 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2524cnveqd 4627 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2625unieqd 3672 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2723, 3, 263eqtr4d 2131 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = {𝐶})
2819, 27jca 301 1 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  Vcvv 2622  {csn 3452  cop 3455   cuni 3661  ccnv 4453  Rel wrel 4459  cfv 5030  1st c1st 5925  2nd c2nd 5926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fo 5036  df-fv 5038  df-1st 5927  df-2nd 5928
This theorem is referenced by:  cnvf1o  6006
  Copyright terms: Public domain W3C validator