Theorem endjusym 7197

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))

Proof of Theorem endjusym

Step	Hyp	Ref	Expression
1		djulf1o 7159	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 5520	. . . . . . . . 9 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ inl:V–1-1→({∅} × V)
4		ssv 3214	. . . . . . . 8 ⊢ 𝐴 ⊆ V
5		f1ores 5536	. . . . . . . 8 ⊢ ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V) → (inl ↾ 𝐴):𝐴–1-1-onto→(inl “ 𝐴))
6	3, 4, 5	mp2an 426	. . . . . . 7 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→(inl “ 𝐴)
7		f1oeng 6847	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (inl ↾ 𝐴):𝐴–1-1-onto→(inl “ 𝐴)) → 𝐴 ≈ (inl “ 𝐴))
8	6, 7	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (inl “ 𝐴))
9	8	ensymd 6874	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
10		djurf1o 7160	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
11		f1of1 5520	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ inr:V–1-1→({1o} × V)
13		f1ores 5536	. . . . . . 7 ⊢ ((inr:V–1-1→({1o} × V) ∧ 𝐴 ⊆ V) → (inr ↾ 𝐴):𝐴–1-1-onto→(inr “ 𝐴))
14	12, 4, 13	mp2an 426	. . . . . 6 ⊢ (inr ↾ 𝐴):𝐴–1-1-onto→(inr “ 𝐴)
15		f1oeng 6847	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (inr ↾ 𝐴):𝐴–1-1-onto→(inr “ 𝐴)) → 𝐴 ≈ (inr “ 𝐴))
16	14, 15	mpan2 425	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (inr “ 𝐴))
17		entr 6875	. . . . 5 ⊢ (((inl “ 𝐴) ≈ 𝐴 ∧ 𝐴 ≈ (inr “ 𝐴)) → (inl “ 𝐴) ≈ (inr “ 𝐴))
18	9, 16, 17	syl2anc 411	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ (inr “ 𝐴))
19	18	adantr 276	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl “ 𝐴) ≈ (inr “ 𝐴))
20		ssv 3214	. . . . . . . 8 ⊢ 𝐵 ⊆ V
21		f1ores 5536	. . . . . . . 8 ⊢ ((inr:V–1-1→({1o} × V) ∧ 𝐵 ⊆ V) → (inr ↾ 𝐵):𝐵–1-1-onto→(inr “ 𝐵))
22	12, 20, 21	mp2an 426	. . . . . . 7 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→(inr “ 𝐵)
23		f1oeng 6847	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ (inr ↾ 𝐵):𝐵–1-1-onto→(inr “ 𝐵)) → 𝐵 ≈ (inr “ 𝐵))
24	22, 23	mpan2 425	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≈ (inr “ 𝐵))
25	24	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≈ (inr “ 𝐵))
26	25	ensymd 6874	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inr “ 𝐵) ≈ 𝐵)
27		f1ores 5536	. . . . . . 7 ⊢ ((inl:V–1-1→({∅} × V) ∧ 𝐵 ⊆ V) → (inl ↾ 𝐵):𝐵–1-1-onto→(inl “ 𝐵))
28	3, 20, 27	mp2an 426	. . . . . 6 ⊢ (inl ↾ 𝐵):𝐵–1-1-onto→(inl “ 𝐵)
29		f1oeng 6847	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ (inl ↾ 𝐵):𝐵–1-1-onto→(inl “ 𝐵)) → 𝐵 ≈ (inl “ 𝐵))
30	28, 29	mpan2 425	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≈ (inl “ 𝐵))
31	30	adantl 277	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≈ (inl “ 𝐵))
32		entr 6875	. . . 4 ⊢ (((inr “ 𝐵) ≈ 𝐵 ∧ 𝐵 ≈ (inl “ 𝐵)) → (inr “ 𝐵) ≈ (inl “ 𝐵))
33	26, 31, 32	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inr “ 𝐵) ≈ (inl “ 𝐵))
34		djuin 7165	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
35	34	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36		incom 3364	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37		djuin 7165	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
38	36, 37	eqtr3i 2227	. . . 4 ⊢ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
39	38	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40		unen 6907	. . 3 ⊢ ((((inl “ 𝐴) ≈ (inr “ 𝐴) ∧ (inr “ 𝐵) ≈ (inl “ 𝐵)) ∧ (((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ ∧ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
41	19, 33, 35, 39, 40	syl22anc 1250	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42		djuun 7168	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
43		uncom 3316	. . 3 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44		djuun 7168	. . 3 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵 ⊔ 𝐴)
45	43, 44	eqtri 2225	. 2 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵 ⊔ 𝐴)
46	41, 42, 45	3brtr3g 4076	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∪ cun 3163   ∩ cin 3164   ⊆ wss 3165  ∅c0 3459  {csn 3632   class class class wbr 4043   × cxp 4672   ↾ cres 4676   “ cima 4677  –1-1→wf1 5267  –1-1-onto→wf1o 5269  1_oc1o 6494   ≈ cen 6824   ⊔ cdju 7138  inlcinl 7146  inrcinr 7147

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-er 6619  df-en 6827  df-dju 7139  df-inl 7148  df-inr 7149

This theorem is referenced by:  sbthom  15898