Theorem endjusym 7400

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 7362	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 5618	. . . . . . . . 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 3264	. . . . . . . 8 ⊢ 𝐴 ⊆ V
5		f1ores 5634	. . . . . . . 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 7009	. . . . . . 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 7036	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
10		djurf1o 7363	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
11		f1of1 5618	. . . . . . . 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 5634	. . . . . . 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 7009	. . . . . 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 7037	. . . . 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 3264	. . . . . . . 8 ⊢ 𝐵 ⊆ V
21		f1ores 5634	. . . . . . . 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 7009	. . . . . . 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 7036	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inr “ 𝐵) ≈ 𝐵)
27		f1ores 5634	. . . . . . 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 7009	. . . . . 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 7037	. . . 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 7368	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
35	34	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36		incom 3415	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37		djuin 7368	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
38	36, 37	eqtr3i 2257	. . . 4 ⊢ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
39	38	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40		unen 7071	. . 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 1275	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42		djuun 7371	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
43		uncom 3367	. . 3 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44		djuun 7371	. . 3 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵 ⊔ 𝐴)
45	43, 44	eqtri 2255	. 2 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵 ⊔ 𝐴)
46	41, 42, 45	3brtr3g 4147	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3212   ∩ cin 3213   ⊆ wss 3214  ∅c0 3512  {csn 3694   class class class wbr 4114   × cxp 4752   ↾ cres 4756   “ cima 4757  –1-1→wf1 5354  –1-1-onto→wf1o 5356  1_oc1o 6653   ≈ cen 6986   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-er 6780  df-en 6989  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  sbthom  16932