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Theorem oacomf1o 8446
Description: Define a bijection from 𝐴 +o 𝐵 to 𝐵 +o 𝐴. Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g., oancom 9487). (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1o.1 𝐹 = ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
Assertion
Ref Expression
oacomf1o ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem oacomf1o
StepHypRef Expression
1 eqid 2737 . . . . . . 7 (𝑥𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥𝐴 ↦ (𝐵 +o 𝑥))
21oacomf1olem 8445 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∧ (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅))
32simpld 495 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)))
4 eqid 2737 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8445 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 459 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 495 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1ocnv 6766 . . . . . 6 ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))–1-1-onto𝐵)
97, 8syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))–1-1-onto𝐵)
10 incom 4146 . . . . . 6 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 496 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11eqtrid 2789 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
132simprd 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14 f1oun 6773 . . . . 5 ((((𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∧ (𝑥𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))–1-1-onto𝐵) ∧ ((𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅ ∧ (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)) → ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
153, 9, 12, 13, 14syl22anc 836 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
16 oacomf1o.1 . . . . 5 𝐹 = ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
17 f1oeq1 6742 . . . . 5 (𝐹 = ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥))) → (𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
1816, 17ax-mp 5 . . . 4 (𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
1915, 18sylibr 233 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
20 oarec 8443 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2120f1oeq2d 6750 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
2219, 21mpbird 256 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23 oarec 8443 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
2423ancoms 459 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
25 uncom 4098 . . . 4 (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
2624, 25eqtrdi 2793 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
2726f1oeq3d 6751 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴) ↔ 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
2822, 27mpbird 256 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  cun 3895  cin 3896  c0 4267  cmpt 5170  ccnv 5607  ran crn 5609  Oncon0 6289  1-1-ontowf1o 6465  (class class class)co 7317   +o coa 8343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-oadd 8350
This theorem is referenced by:  cnfcomlem  9535
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