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Theorem cnvf1olem 7801
 Description: Lemma for cnvf1o 7802. (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 772 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {𝐵})
2 1st2nd 7733 . . . . . . . . 9 ((Rel 𝐴𝐵𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
32adantrr 716 . . . . . . . 8 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
43sneqd 4540 . . . . . . 7 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
54cnveqd 5714 . . . . . 6 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
65unieqd 4818 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
71, 6eqtrd 2833 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {⟨(1st𝐵), (2nd𝐵)⟩})
8 opswap 6057 . . . 4 {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩
97, 8eqtrdi 2849 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = ⟨(2nd𝐵), (1st𝐵)⟩)
10 simprl 770 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵𝐴)
113, 10eqeltrrd 2891 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
12 fvex 6668 . . . . 5 (2nd𝐵) ∈ V
13 fvex 6668 . . . . 5 (1st𝐵) ∈ V
1412, 13opelcnv 5720 . . . 4 (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
1511, 14sylibr 237 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴)
169, 15eqeltrd 2890 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶𝐴)
17 opswap 6057 . . . 4 {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩
1817eqcomi 2807 . . 3 ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩}
199sneqd 4540 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2019cnveqd 5714 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2120unieqd 4818 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2218, 3, 213eqtr4a 2859 . 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 1538   ∈ wcel 2111  {csn 4528  ⟨cop 4534  ∪ cuni 4804  ◡ccnv 5522  Rel wrel 5528  ‘cfv 6332  1st c1st 7682  2nd c2nd 7683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fv 6340  df-1st 7684  df-2nd 7685 This theorem is referenced by:  cnvf1o  7802  fcnvgreu  30480  gsumhashmul  30790
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