MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oacomf1o Structured version   Visualization version   GIF version

Theorem oacomf1o 8513
Description: Define a bijection from 𝐴 +o 𝐡 to 𝐡 +o 𝐴. Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g., oancom 9592). (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1o.1 𝐹 = ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
Assertion
Ref Expression
oacomf1o ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(𝐡 +o 𝐴))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem oacomf1o
StepHypRef Expression
1 eqid 2733 . . . . . . 7 (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) = (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))
21oacomf1olem 8512 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)):𝐴–1-1-ontoβ†’ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) ∧ (ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) ∩ 𝐡) = βˆ…))
32simpld 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)):𝐴–1-1-ontoβ†’ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)))
4 eqid 2733 . . . . . . . . 9 (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) = (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))
54oacomf1olem 8512 . . . . . . . 8 ((𝐡 ∈ On ∧ 𝐴 ∈ On) β†’ ((π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∧ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…))
65ancoms 460 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∧ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…))
76simpld 496 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
8 f1ocnv 6797 . . . . . 6 ((π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) β†’ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))–1-1-onto→𝐡)
97, 8syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))–1-1-onto→𝐡)
10 incom 4162 . . . . . 6 (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴)
116simprd 497 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…)
1210, 11eqtrid 2785 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ…)
132simprd 497 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) ∩ 𝐡) = βˆ…)
14 f1oun 6804 . . . . 5 ((((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)):𝐴–1-1-ontoβ†’ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) ∧ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))–1-1-onto→𝐡) ∧ ((𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ… ∧ (ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) ∩ 𝐡) = βˆ…)) β†’ ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))):(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
153, 9, 12, 13, 14syl22anc 838 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))):(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
16 oacomf1o.1 . . . . 5 𝐹 = ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
17 f1oeq1 6773 . . . . 5 (𝐹 = ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) β†’ (𝐹:(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡) ↔ ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))):(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡)))
1816, 17ax-mp 5 . . . 4 (𝐹:(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡) ↔ ((π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ β—‘(π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))):(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
1915, 18sylibr 233 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
20 oarec 8510 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) = (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
2120f1oeq2d 6781 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡) ↔ 𝐹:(𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡)))
2219, 21mpbird 257 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
23 oarec 8510 . . . . 5 ((𝐡 ∈ On ∧ 𝐴 ∈ On) β†’ (𝐡 +o 𝐴) = (𝐡 βˆͺ ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))))
2423ancoms 460 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐡 +o 𝐴) = (𝐡 βˆͺ ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))))
25 uncom 4114 . . . 4 (𝐡 βˆͺ ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))) = (ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡)
2624, 25eqtrdi 2789 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐡 +o 𝐴) = (ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡))
2726f1oeq3d 6782 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(𝐡 +o 𝐴) ↔ 𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(ran (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)) βˆͺ 𝐡)))
2822, 27mpbird 257 1 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:(𝐴 +o 𝐡)–1-1-ontoβ†’(𝐡 +o 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3909   ∩ cin 3910  βˆ…c0 4283   ↦ cmpt 5189  β—‘ccnv 5633  ran crn 5635  Oncon0 6318  β€“1-1-ontoβ†’wf1o 6496  (class class class)co 7358   +o coa 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-oadd 8417
This theorem is referenced by:  cnfcomlem  9640
  Copyright terms: Public domain W3C validator