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Theorem cnvf1olem 5873
Description: Lemma for cnvf1o 5874. (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 492 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {𝐵})
2 1st2nd 5835 . . . . . . . 8 ((Rel 𝐴𝐵𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
32adantrr 456 . . . . . . 7 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
43sneqd 3416 . . . . . 6 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
54cnveqd 4539 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
65unieqd 3619 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
7 1stexg 5822 . . . . . 6 (𝐵𝐴 → (1st𝐵) ∈ V)
8 2ndexg 5823 . . . . . 6 (𝐵𝐴 → (2nd𝐵) ∈ V)
9 opswapg 4835 . . . . . 6 (((1st𝐵) ∈ V ∧ (2nd𝐵) ∈ V) → {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
107, 8, 9syl2anc 397 . . . . 5 (𝐵𝐴 {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
1110ad2antrl 467 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩)
121, 6, 113eqtrd 2092 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = ⟨(2nd𝐵), (1st𝐵)⟩)
13 simprl 491 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵𝐴)
143, 13eqeltrrd 2131 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
15 opelcnvg 4543 . . . . . 6 (((2nd𝐵) ∈ V ∧ (1st𝐵) ∈ V) → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
168, 7, 15syl2anc 397 . . . . 5 (𝐵𝐴 → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
1716ad2antrl 467 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴))
1814, 17mpbird 160 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴)
1912, 18eqeltrd 2130 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶𝐴)
20 opswapg 4835 . . . . . 6 (((2nd𝐵) ∈ V ∧ (1st𝐵) ∈ V) → {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩)
218, 7, 20syl2anc 397 . . . . 5 (𝐵𝐴 {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩)
2221eqcomd 2061 . . . 4 (𝐵𝐴 → ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩})
2322ad2antrl 467 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩})
2412sneqd 3416 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2524cnveqd 4539 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2625unieqd 3619 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2723, 3, 263eqtr4d 2098 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = {𝐶})
2819, 27jca 294 1 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406   cuni 3608  ccnv 4372  Rel wrel 4378  cfv 4930  1st c1st 5793  2nd c2nd 5794
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  cnvf1o  5874
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