MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvf1olem Structured version   Visualization version   GIF version

Theorem cnvf1olem 7878
Description: Lemma for cnvf1o 7879. (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 773 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {𝐵})
2 1st2nd 7810 . . . . . . . . 9 ((Rel 𝐴𝐵𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
32adantrr 717 . . . . . . . 8 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
43sneqd 4553 . . . . . . 7 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
54cnveqd 5744 . . . . . 6 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
65unieqd 4833 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
71, 6eqtrd 2777 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {⟨(1st𝐵), (2nd𝐵)⟩})
8 opswap 6092 . . . 4 {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩
97, 8eqtrdi 2794 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = ⟨(2nd𝐵), (1st𝐵)⟩)
10 simprl 771 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵𝐴)
113, 10eqeltrrd 2839 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
12 fvex 6730 . . . . 5 (2nd𝐵) ∈ V
13 fvex 6730 . . . . 5 (1st𝐵) ∈ V
1412, 13opelcnv 5750 . . . 4 (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
1511, 14sylibr 237 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴)
169, 15eqeltrd 2838 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶𝐴)
17 opswap 6092 . . . 4 {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩
1817eqcomi 2746 . . 3 ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩}
199sneqd 4553 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2019cnveqd 5744 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2120unieqd 4833 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2218, 3, 213eqtr4a 2804 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = {𝐶})
2316, 22jca 515 1 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  {csn 4541  cop 4547   cuni 4819  ccnv 5550  Rel wrel 5556  cfv 6380  1st c1st 7759  2nd c2nd 7760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fv 6388  df-1st 7761  df-2nd 7762
This theorem is referenced by:  cnvf1o  7879  fcnvgreu  30730  gsumhashmul  31035
  Copyright terms: Public domain W3C validator