Theorem endjusym 7097

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 7059	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 5462	. . . . . . . . 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 3179	. . . . . . . 8 ⊢ 𝐴 ⊆ V
5		f1ores 5478	. . . . . . . 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 6759	. . . . . . 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 6785	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
10		djurf1o 7060	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
11		f1of1 5462	. . . . . . . 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 5478	. . . . . . 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 6759	. . . . . 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 6786	. . . . 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 3179	. . . . . . . 8 ⊢ 𝐵 ⊆ V
21		f1ores 5478	. . . . . . . 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 6759	. . . . . . 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 6785	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inr “ 𝐵) ≈ 𝐵)
27		f1ores 5478	. . . . . . 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 6759	. . . . . 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 6786	. . . 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 7065	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
35	34	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36		incom 3329	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37		djuin 7065	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
38	36, 37	eqtr3i 2200	. . . 4 ⊢ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
39	38	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40		unen 6818	. . 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 1239	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42		djuun 7068	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
43		uncom 3281	. . 3 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44		djuun 7068	. . 3 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵 ⊔ 𝐴)
45	43, 44	eqtri 2198	. 2 ⊢ ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵 ⊔ 𝐴)
46	41, 42, 45	3brtr3g 4038	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∪ cun 3129   ∩ cin 3130   ⊆ wss 3131  ∅c0 3424  {csn 3594   class class class wbr 4005   × cxp 4626   ↾ cres 4630   “ cima 4631  –1-1→wf1 5215  –1-1-onto→wf1o 5217  1_oc1o 6412   ≈ cen 6740   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-er 6537  df-en 6743  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  sbthom  14859