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Theorem f1oprg 6219
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6218. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
f1oprg (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))

Proof of Theorem f1oprg
StepHypRef Expression
1 f1osng 6215 . . . . 5 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
21ad2antrr 762 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
3 f1osng 6215 . . . . 5 ((𝐶𝑋𝐷𝑌) → {⟨𝐶, 𝐷⟩}:{𝐶}–1-1-onto→{𝐷})
43ad2antlr 763 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → {⟨𝐶, 𝐷⟩}:{𝐶}–1-1-onto→{𝐷})
5 disjsn2 4279 . . . . 5 (𝐴𝐶 → ({𝐴} ∩ {𝐶}) = ∅)
65ad2antrl 764 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({𝐴} ∩ {𝐶}) = ∅)
7 disjsn2 4279 . . . . 5 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
87ad2antll 765 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({𝐵} ∩ {𝐷}) = ∅)
9 f1oun 6194 . . . 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 1367 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}):({𝐴} ∪ {𝐶})–1-1-onto→({𝐵} ∪ {𝐷}))
11 df-pr 4213 . . . . . 6 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
1211eqcomi 2660 . . . . 5 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
1312a1i 11 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
14 df-pr 4213 . . . . . 6 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
1514eqcomi 2660 . . . . 5 ({𝐴} ∪ {𝐶}) = {𝐴, 𝐶}
1615a1i 11 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({𝐴} ∪ {𝐶}) = {𝐴, 𝐶})
17 df-pr 4213 . . . . . 6 {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
1817eqcomi 2660 . . . . 5 ({𝐵} ∪ {𝐷}) = {𝐵, 𝐷}
1918a1i 11 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → ({𝐵} ∪ {𝐷}) = {𝐵, 𝐷})
2013, 16, 19f1oeq123d 6171 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → (({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}):({𝐴} ∪ {𝐶})–1-1-onto→({𝐵} ∪ {𝐷}) ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))
2110, 20mpbid 222 . 2 ((((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) ∧ (𝐴𝐶𝐵𝐷)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷})
2221ex 449 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212  cop 4216  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by:  f1prex  6579  s2f1o  13707  f1oun2prg  13708  symg2bas  17864  poimirlem9  33548  poimirlem15  33554
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