Theorem endjusym 7286

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 7248	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 5579	. . . . . . . . 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 3247	. . . . . . . 8 ⊢ 𝐴 ⊆ V
5		f1ores 5595	. . . . . . . 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 6925	. . . . . . 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 6952	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
10		djurf1o 7249	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
11		f1of1 5579	. . . . . . . 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 5595	. . . . . . 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 6925	. . . . . 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 6953	. . . . 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 3247	. . . . . . . 8 ⊢ 𝐵 ⊆ V
21		f1ores 5595	. . . . . . . 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 6925	. . . . . . 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 6952	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inr “ 𝐵) ≈ 𝐵)
27		f1ores 5595	. . . . . . 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 6925	. . . . . 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 6953	. . . 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 7254	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
35	34	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36		incom 3397	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37		djuin 7254	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
38	36, 37	eqtr3i 2252	. . . 4 ⊢ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
39	38	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40		unen 6986	. . 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 1272	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42		djuun 7257	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
43		uncom 3349	. . 3 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44		djuun 7257	. . 3 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵 ⊔ 𝐴)
45	43, 44	eqtri 2250	. 2 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵 ⊔ 𝐴)
46	41, 42, 45	3brtr3g 4119	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  {csn 3667   class class class wbr 4086   × cxp 4721   ↾ cres 4725   “ cima 4726  –1-1→wf1 5321  –1-1-onto→wf1o 5323  1_oc1o 6570   ≈ cen 6902   ⊔ cdju 7227  inlcinl 7235  inrcinr 7236

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-1o 6577  df-er 6697  df-en 6905  df-dju 7228  df-inl 7237  df-inr 7238

This theorem is referenced by:  sbthom  16566