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Theorem cnvf1olem 7430
Description: Lemma for cnvf1o 7431. (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 756 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {𝐵})
2 1st2nd 7367 . . . . . . . . 9 ((Rel 𝐴𝐵𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
32adantrr 696 . . . . . . . 8 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
43sneqd 4329 . . . . . . 7 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
54cnveqd 5435 . . . . . 6 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
65unieqd 4585 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐵} = {⟨(1st𝐵), (2nd𝐵)⟩})
71, 6eqtrd 2805 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = {⟨(1st𝐵), (2nd𝐵)⟩})
8 opswap 5765 . . . 4 {⟨(1st𝐵), (2nd𝐵)⟩} = ⟨(2nd𝐵), (1st𝐵)⟩
97, 8syl6eq 2821 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶 = ⟨(2nd𝐵), (1st𝐵)⟩)
10 simprl 754 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵𝐴)
113, 10eqeltrrd 2851 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
12 fvex 6344 . . . . 5 (2nd𝐵) ∈ V
13 fvex 6344 . . . . 5 (1st𝐵) ∈ V
1412, 13opelcnv 5441 . . . 4 (⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴 ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ 𝐴)
1511, 14sylibr 224 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → ⟨(2nd𝐵), (1st𝐵)⟩ ∈ 𝐴)
169, 15eqeltrd 2850 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐶𝐴)
17 opswap 5765 . . . 4 {⟨(2nd𝐵), (1st𝐵)⟩} = ⟨(1st𝐵), (2nd𝐵)⟩
1817eqcomi 2780 . . 3 ⟨(1st𝐵), (2nd𝐵)⟩ = {⟨(2nd𝐵), (1st𝐵)⟩}
199sneqd 4329 . . . . 5 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2019cnveqd 5435 . . . 4 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2120unieqd 4585 . . 3 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → {𝐶} = {⟨(2nd𝐵), (1st𝐵)⟩})
2218, 3, 213eqtr4a 2831 . 2 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → 𝐵 = {𝐶})
2316, 22jca 501 1 ((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {csn 4317  cop 4323   cuni 4575  ccnv 5249  Rel wrel 5255  cfv 6030  1st c1st 7317  2nd c2nd 7318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-1st 7319  df-2nd 7320
This theorem is referenced by:  cnvf1o  7431  fcnvgreu  29812
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