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Theorem oacomf1o 8621
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 9720). (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 2740 . . . . . . 7 (𝑥𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥𝐴 ↦ (𝐵 +o 𝑥))
21oacomf1olem 8620 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∧ (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅))
32simpld 494 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐴 ↦ (𝐵 +o 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)))
4 eqid 2740 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8620 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 458 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 494 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1ocnv 6874 . . . . . 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 4230 . . . . . 6 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 495 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11eqtrid 2792 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
132simprd 495 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14 f1oun 6881 . . . . 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 838 . . . 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 6850 . . . . 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 234 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
20 oarec 8618 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2120f1oeq2d 6858 . . 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 257 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23 oarec 8618 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
2423ancoms 458 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))))
25 uncom 4181 . . . 4 (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
2624, 25eqtrdi 2796 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
2726f1oeq3d 6859 . 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 257 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 206  wa 395   = wceq 1537  wcel 2108  cun 3974  cin 3975  c0 4352  cmpt 5249  ccnv 5699  ran crn 5701  Oncon0 6395  1-1-ontowf1o 6572  (class class class)co 7448   +o coa 8519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526
This theorem is referenced by:  cnfcomlem  9768
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