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Theorem oacomf1o 8187
 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 9111). (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 2824 . . . . . . 7 (𝑥𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥𝐴 ↦ (𝐵 +o 𝑥))
21oacomf1olem 8186 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∧ (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅))
32simpld 498 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)))
4 eqid 2824 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8186 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 462 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 498 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1ocnv 6618 . . . . . 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 4163 . . . . . 6 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 499 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11syl5eq 2871 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
132simprd 499 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14 f1oun 6625 . . . . 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 837 . . . 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 6595 . . . . 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 237 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
20 oarec 8184 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2120f1oeq2d 6602 . . 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 260 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23 oarec 8184 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
2423ancoms 462 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
25 uncom 4115 . . . 4 (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
2624, 25syl6eq 2875 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
2726f1oeq3d 6603 . 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 260 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ∪ cun 3917   ∩ cin 3918  ∅c0 4276   ↦ cmpt 5132  ◡ccnv 5541  ran crn 5543  Oncon0 6178  –1-1-onto→wf1o 6342  (class class class)co 7149   +o coa 8095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102 This theorem is referenced by:  cnfcomlem  9159
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