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Theorem cnvf1olem 5768
Description: Lemma for cnvf1o 5769. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1olem ((Rel A (B A 𝐶 = {B})) → (𝐶 A B = {𝐶}))

Proof of Theorem cnvf1olem
StepHypRef Expression
1 simprr 472 . . . 4 ((Rel A (B A 𝐶 = {B})) → 𝐶 = {B})
2 1st2nd 5730 . . . . . . . 8 ((Rel A B A) → B = ⟨(1stB), (2ndB)⟩)
32adantrr 451 . . . . . . 7 ((Rel A (B A 𝐶 = {B})) → B = ⟨(1stB), (2ndB)⟩)
43sneqd 3363 . . . . . 6 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
54cnveqd 4438 . . . . 5 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
65unieqd 3565 . . . 4 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
7 1stexg 5717 . . . . . 6 (B A → (1stB) V)
8 2ndexg 5718 . . . . . 6 (B A → (2ndB) V)
9 opswapg 4734 . . . . . 6 (((1stB) V (2ndB) V) → {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
107, 8, 9syl2anc 393 . . . . 5 (B A {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
1110ad2antrl 463 . . . 4 ((Rel A (B A 𝐶 = {B})) → {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
121, 6, 113eqtrd 2058 . . 3 ((Rel A (B A 𝐶 = {B})) → 𝐶 = ⟨(2ndB), (1stB)⟩)
13 simprl 471 . . . . 5 ((Rel A (B A 𝐶 = {B})) → B A)
143, 13eqeltrrd 2097 . . . 4 ((Rel A (B A 𝐶 = {B})) → ⟨(1stB), (2ndB)⟩ A)
15 opelcnvg 4442 . . . . . 6 (((2ndB) V (1stB) V) → (⟨(2ndB), (1stB)⟩ A ↔ ⟨(1stB), (2ndB)⟩ A))
168, 7, 15syl2anc 393 . . . . 5 (B A → (⟨(2ndB), (1stB)⟩ A ↔ ⟨(1stB), (2ndB)⟩ A))
1716ad2antrl 463 . . . 4 ((Rel A (B A 𝐶 = {B})) → (⟨(2ndB), (1stB)⟩ A ↔ ⟨(1stB), (2ndB)⟩ A))
1814, 17mpbird 156 . . 3 ((Rel A (B A 𝐶 = {B})) → ⟨(2ndB), (1stB)⟩ A)
1912, 18eqeltrd 2096 . 2 ((Rel A (B A 𝐶 = {B})) → 𝐶 A)
20 opswapg 4734 . . . . . 6 (((2ndB) V (1stB) V) → {⟨(2ndB), (1stB)⟩} = ⟨(1stB), (2ndB)⟩)
218, 7, 20syl2anc 393 . . . . 5 (B A {⟨(2ndB), (1stB)⟩} = ⟨(1stB), (2ndB)⟩)
2221eqcomd 2027 . . . 4 (B A → ⟨(1stB), (2ndB)⟩ = {⟨(2ndB), (1stB)⟩})
2322ad2antrl 463 . . 3 ((Rel A (B A 𝐶 = {B})) → ⟨(1stB), (2ndB)⟩ = {⟨(2ndB), (1stB)⟩})
2412sneqd 3363 . . . . 5 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2524cnveqd 4438 . . . 4 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2625unieqd 3565 . . 3 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2723, 3, 263eqtr4d 2064 . 2 ((Rel A (B A 𝐶 = {B})) → B = {𝐶})
2819, 27jca 290 1 ((Rel A (B A 𝐶 = {B})) → (𝐶 A B = {𝐶}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350  cop 3353   cuni 3554  ccnv 4271  Rel wrel 4277  cfv 4829  1st c1st 5688  2nd c2nd 5689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by:  cnvf1o  5769
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