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Theorem cnvf1olem 5787
Description: Lemma for cnvf1o 5788. (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 484 . . . 4 ((Rel A (B A 𝐶 = {B})) → 𝐶 = {B})
2 1st2nd 5749 . . . . . . . 8 ((Rel A B A) → B = ⟨(1stB), (2ndB)⟩)
32adantrr 448 . . . . . . 7 ((Rel A (B A 𝐶 = {B})) → B = ⟨(1stB), (2ndB)⟩)
43sneqd 3380 . . . . . 6 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
54cnveqd 4454 . . . . 5 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
65unieqd 3582 . . . 4 ((Rel A (B A 𝐶 = {B})) → {B} = {⟨(1stB), (2ndB)⟩})
7 1stexg 5736 . . . . . 6 (B A → (1stB) V)
8 2ndexg 5737 . . . . . 6 (B A → (2ndB) V)
9 opswapg 4750 . . . . . 6 (((1stB) V (2ndB) V) → {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
107, 8, 9syl2anc 391 . . . . 5 (B A {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
1110ad2antrl 459 . . . 4 ((Rel A (B A 𝐶 = {B})) → {⟨(1stB), (2ndB)⟩} = ⟨(2ndB), (1stB)⟩)
121, 6, 113eqtrd 2073 . . 3 ((Rel A (B A 𝐶 = {B})) → 𝐶 = ⟨(2ndB), (1stB)⟩)
13 simprl 483 . . . . 5 ((Rel A (B A 𝐶 = {B})) → B A)
143, 13eqeltrrd 2112 . . . 4 ((Rel A (B A 𝐶 = {B})) → ⟨(1stB), (2ndB)⟩ A)
15 opelcnvg 4458 . . . . . 6 (((2ndB) V (1stB) V) → (⟨(2ndB), (1stB)⟩ A ↔ ⟨(1stB), (2ndB)⟩ A))
168, 7, 15syl2anc 391 . . . . 5 (B A → (⟨(2ndB), (1stB)⟩ A ↔ ⟨(1stB), (2ndB)⟩ A))
1716ad2antrl 459 . . . 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 2111 . 2 ((Rel A (B A 𝐶 = {B})) → 𝐶 A)
20 opswapg 4750 . . . . . 6 (((2ndB) V (1stB) V) → {⟨(2ndB), (1stB)⟩} = ⟨(1stB), (2ndB)⟩)
218, 7, 20syl2anc 391 . . . . 5 (B A {⟨(2ndB), (1stB)⟩} = ⟨(1stB), (2ndB)⟩)
2221eqcomd 2042 . . . 4 (B A → ⟨(1stB), (2ndB)⟩ = {⟨(2ndB), (1stB)⟩})
2322ad2antrl 459 . . 3 ((Rel A (B A 𝐶 = {B})) → ⟨(1stB), (2ndB)⟩ = {⟨(2ndB), (1stB)⟩})
2412sneqd 3380 . . . . 5 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2524cnveqd 4454 . . . 4 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2625unieqd 3582 . . 3 ((Rel A (B A 𝐶 = {B})) → {𝐶} = {⟨(2ndB), (1stB)⟩})
2723, 3, 263eqtr4d 2079 . 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 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cuni 3571  ccnv 4287  Rel wrel 4293  cfv 4845  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  cnvf1o  5788
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