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Theorem endjusym 7095
Description: Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
Assertion
Ref Expression
endjusym ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (š“ āŠ” šµ) ā‰ˆ (šµ āŠ” š“))

Proof of Theorem endjusym
StepHypRef Expression
1 djulf1o 7057 . . . . . . . . 9 inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V)
2 f1of1 5461 . . . . . . . . 9 (inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V) ā†’ inl:Vā€“1-1ā†’({āˆ…} Ɨ V))
31, 2ax-mp 5 . . . . . . . 8 inl:Vā€“1-1ā†’({āˆ…} Ɨ V)
4 ssv 3178 . . . . . . . 8 š“ āŠ† V
5 f1ores 5477 . . . . . . . 8 ((inl:Vā€“1-1ā†’({āˆ…} Ɨ V) āˆ§ š“ āŠ† V) ā†’ (inl ā†¾ š“):š“ā€“1-1-ontoā†’(inl ā€œ š“))
63, 4, 5mp2an 426 . . . . . . 7 (inl ā†¾ š“):š“ā€“1-1-ontoā†’(inl ā€œ š“)
7 f1oeng 6757 . . . . . . 7 ((š“ āˆˆ š‘‰ āˆ§ (inl ā†¾ š“):š“ā€“1-1-ontoā†’(inl ā€œ š“)) ā†’ š“ ā‰ˆ (inl ā€œ š“))
86, 7mpan2 425 . . . . . 6 (š“ āˆˆ š‘‰ ā†’ š“ ā‰ˆ (inl ā€œ š“))
98ensymd 6783 . . . . 5 (š“ āˆˆ š‘‰ ā†’ (inl ā€œ š“) ā‰ˆ š“)
10 djurf1o 7058 . . . . . . . 8 inr:Vā€“1-1-ontoā†’({1o} Ɨ V)
11 f1of1 5461 . . . . . . . 8 (inr:Vā€“1-1-ontoā†’({1o} Ɨ V) ā†’ inr:Vā€“1-1ā†’({1o} Ɨ V))
1210, 11ax-mp 5 . . . . . . 7 inr:Vā€“1-1ā†’({1o} Ɨ V)
13 f1ores 5477 . . . . . . 7 ((inr:Vā€“1-1ā†’({1o} Ɨ V) āˆ§ š“ āŠ† V) ā†’ (inr ā†¾ š“):š“ā€“1-1-ontoā†’(inr ā€œ š“))
1412, 4, 13mp2an 426 . . . . . 6 (inr ā†¾ š“):š“ā€“1-1-ontoā†’(inr ā€œ š“)
15 f1oeng 6757 . . . . . 6 ((š“ āˆˆ š‘‰ āˆ§ (inr ā†¾ š“):š“ā€“1-1-ontoā†’(inr ā€œ š“)) ā†’ š“ ā‰ˆ (inr ā€œ š“))
1614, 15mpan2 425 . . . . 5 (š“ āˆˆ š‘‰ ā†’ š“ ā‰ˆ (inr ā€œ š“))
17 entr 6784 . . . . 5 (((inl ā€œ š“) ā‰ˆ š“ āˆ§ š“ ā‰ˆ (inr ā€œ š“)) ā†’ (inl ā€œ š“) ā‰ˆ (inr ā€œ š“))
189, 16, 17syl2anc 411 . . . 4 (š“ āˆˆ š‘‰ ā†’ (inl ā€œ š“) ā‰ˆ (inr ā€œ š“))
1918adantr 276 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (inl ā€œ š“) ā‰ˆ (inr ā€œ š“))
20 ssv 3178 . . . . . . . 8 šµ āŠ† V
21 f1ores 5477 . . . . . . . 8 ((inr:Vā€“1-1ā†’({1o} Ɨ V) āˆ§ šµ āŠ† V) ā†’ (inr ā†¾ šµ):šµā€“1-1-ontoā†’(inr ā€œ šµ))
2212, 20, 21mp2an 426 . . . . . . 7 (inr ā†¾ šµ):šµā€“1-1-ontoā†’(inr ā€œ šµ)
23 f1oeng 6757 . . . . . . 7 ((šµ āˆˆ š‘Š āˆ§ (inr ā†¾ šµ):šµā€“1-1-ontoā†’(inr ā€œ šµ)) ā†’ šµ ā‰ˆ (inr ā€œ šµ))
2422, 23mpan2 425 . . . . . 6 (šµ āˆˆ š‘Š ā†’ šµ ā‰ˆ (inr ā€œ šµ))
2524adantl 277 . . . . 5 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ šµ ā‰ˆ (inr ā€œ šµ))
2625ensymd 6783 . . . 4 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (inr ā€œ šµ) ā‰ˆ šµ)
27 f1ores 5477 . . . . . . 7 ((inl:Vā€“1-1ā†’({āˆ…} Ɨ V) āˆ§ šµ āŠ† V) ā†’ (inl ā†¾ šµ):šµā€“1-1-ontoā†’(inl ā€œ šµ))
283, 20, 27mp2an 426 . . . . . 6 (inl ā†¾ šµ):šµā€“1-1-ontoā†’(inl ā€œ šµ)
29 f1oeng 6757 . . . . . 6 ((šµ āˆˆ š‘Š āˆ§ (inl ā†¾ šµ):šµā€“1-1-ontoā†’(inl ā€œ šµ)) ā†’ šµ ā‰ˆ (inl ā€œ šµ))
3028, 29mpan2 425 . . . . 5 (šµ āˆˆ š‘Š ā†’ šµ ā‰ˆ (inl ā€œ šµ))
3130adantl 277 . . . 4 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ šµ ā‰ˆ (inl ā€œ šµ))
32 entr 6784 . . . 4 (((inr ā€œ šµ) ā‰ˆ šµ āˆ§ šµ ā‰ˆ (inl ā€œ šµ)) ā†’ (inr ā€œ šµ) ā‰ˆ (inl ā€œ šµ))
3326, 31, 32syl2anc 411 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (inr ā€œ šµ) ā‰ˆ (inl ā€œ šµ))
34 djuin 7063 . . . 4 ((inl ā€œ š“) āˆ© (inr ā€œ šµ)) = āˆ…
3534a1i 9 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ ((inl ā€œ š“) āˆ© (inr ā€œ šµ)) = āˆ…)
36 incom 3328 . . . . 5 ((inl ā€œ šµ) āˆ© (inr ā€œ š“)) = ((inr ā€œ š“) āˆ© (inl ā€œ šµ))
37 djuin 7063 . . . . 5 ((inl ā€œ šµ) āˆ© (inr ā€œ š“)) = āˆ…
3836, 37eqtr3i 2200 . . . 4 ((inr ā€œ š“) āˆ© (inl ā€œ šµ)) = āˆ…
3938a1i 9 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ ((inr ā€œ š“) āˆ© (inl ā€œ šµ)) = āˆ…)
40 unen 6816 . . 3 ((((inl ā€œ š“) ā‰ˆ (inr ā€œ š“) āˆ§ (inr ā€œ šµ) ā‰ˆ (inl ā€œ šµ)) āˆ§ (((inl ā€œ š“) āˆ© (inr ā€œ šµ)) = āˆ… āˆ§ ((inr ā€œ š“) āˆ© (inl ā€œ šµ)) = āˆ…)) ā†’ ((inl ā€œ š“) āˆŖ (inr ā€œ šµ)) ā‰ˆ ((inr ā€œ š“) āˆŖ (inl ā€œ šµ)))
4119, 33, 35, 39, 40syl22anc 1239 . 2 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ ((inl ā€œ š“) āˆŖ (inr ā€œ šµ)) ā‰ˆ ((inr ā€œ š“) āˆŖ (inl ā€œ šµ)))
42 djuun 7066 . 2 ((inl ā€œ š“) āˆŖ (inr ā€œ šµ)) = (š“ āŠ” šµ)
43 uncom 3280 . . 3 ((inr ā€œ š“) āˆŖ (inl ā€œ šµ)) = ((inl ā€œ šµ) āˆŖ (inr ā€œ š“))
44 djuun 7066 . . 3 ((inl ā€œ šµ) āˆŖ (inr ā€œ š“)) = (šµ āŠ” š“)
4543, 44eqtri 2198 . 2 ((inr ā€œ š“) āˆŖ (inl ā€œ šµ)) = (šµ āŠ” š“)
4641, 42, 453brtr3g 4037 1 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (š“ āŠ” šµ) ā‰ˆ (šµ āŠ” š“))
Colors of variables: wff set class
Syntax hints:   ā†’ wi 4   āˆ§ wa 104   = wceq 1353   āˆˆ wcel 2148  Vcvv 2738   āˆŖ cun 3128   āˆ© cin 3129   āŠ† wss 3130  āˆ…c0 3423  {csn 3593   class class class wbr 4004   Ɨ cxp 4625   ā†¾ cres 4629   ā€œ cima 4630  ā€“1-1ā†’wf1 5214  ā€“1-1-ontoā†’wf1o 5216  1oc1o 6410   ā‰ˆ cen 6738   āŠ” cdju 7036  inlcinl 7044  inrcinr 7045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-er 6535  df-en 6741  df-dju 7037  df-inl 7046  df-inr 7047
This theorem is referenced by:  sbthom  14777
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