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Theorem f1oprg 5111
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
f1oprg (((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ((A𝐶 B𝐷) → {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷}))

Proof of Theorem f1oprg
StepHypRef Expression
1 f1osng 5110 . . . . 5 ((A 𝑉 B 𝑊) → {⟨A, B⟩}:{A}–1-1-onto→{B})
21ad2antrr 457 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → {⟨A, B⟩}:{A}–1-1-onto→{B})
3 f1osng 5110 . . . . 5 ((𝐶 𝑋 𝐷 𝑌) → {⟨𝐶, 𝐷⟩}:{𝐶}–1-1-onto→{𝐷})
43ad2antlr 458 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → {⟨𝐶, 𝐷⟩}:{𝐶}–1-1-onto→{𝐷})
5 disjsn2 3424 . . . . 5 (A𝐶 → ({A} ∩ {𝐶}) = ∅)
65ad2antrl 459 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({A} ∩ {𝐶}) = ∅)
7 disjsn2 3424 . . . . 5 (B𝐷 → ({B} ∩ {𝐷}) = ∅)
87ad2antll 460 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({B} ∩ {𝐷}) = ∅)
9 f1oun 5089 . . . 4 ((({⟨A, B⟩}:{A}–1-1-onto→{B} {⟨𝐶, 𝐷⟩}:{𝐶}–1-1-onto→{𝐷}) (({A} ∩ {𝐶}) = ∅ ({B} ∩ {𝐷}) = ∅)) → ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}):({A} ∪ {𝐶})–1-1-onto→({B} ∪ {𝐷}))
102, 4, 6, 8, 9syl22anc 1135 . . 3 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}):({A} ∪ {𝐶})–1-1-onto→({B} ∪ {𝐷}))
11 df-pr 3374 . . . . . 6 {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩})
1211eqcomi 2041 . . . . 5 ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}) = {⟨A, B⟩, ⟨𝐶, 𝐷⟩}
1312a1i 9 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}) = {⟨A, B⟩, ⟨𝐶, 𝐷⟩})
14 df-pr 3374 . . . . . 6 {A, 𝐶} = ({A} ∪ {𝐶})
1514eqcomi 2041 . . . . 5 ({A} ∪ {𝐶}) = {A, 𝐶}
1615a1i 9 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({A} ∪ {𝐶}) = {A, 𝐶})
17 df-pr 3374 . . . . . 6 {B, 𝐷} = ({B} ∪ {𝐷})
1817eqcomi 2041 . . . . 5 ({B} ∪ {𝐷}) = {B, 𝐷}
1918a1i 9 . . . 4 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → ({B} ∪ {𝐷}) = {B, 𝐷})
2013, 16, 19f1oeq123d 5066 . . 3 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → (({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}):({A} ∪ {𝐶})–1-1-onto→({B} ∪ {𝐷}) ↔ {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷}))
2110, 20mpbid 135 . 2 ((((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) (A𝐶 B𝐷)) → {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷})
2221ex 108 1 (((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ((A𝐶 B𝐷) → {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wne 2201  cun 2909  cin 2910  c0 3218  {csn 3367  {cpr 3368  cop 3370  1-1-ontowf1o 4844
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-in1 544  ax-in2 545  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-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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by: (None)
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