Theorem eusvobj2 5807

Description: Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
eusvobj1.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eusvobj2	⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusvobj2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3628	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧})
2		eleq2 2221	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3		abid 2145	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		velsn 3577	. . . . . 6 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
5	2, 3, 4	3bitr3g 221	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ 𝑥 = 𝑧))
6		nfre1 2500	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵
7	6	nfab 2304	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵}
8	7	nfeq1 2309	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧}
9		eusvobj1.1	. . . . . . . . 9 ⊢ 𝐵 ∈ V
10	9	elabrex 5705	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵})
11		eleq2 2221	. . . . . . . . 9 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
12	9	elsn 3576	. . . . . . . . . 10 ⊢ (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13		eqcom 2159	. . . . . . . . . 10 ⊢ (𝐵 = 𝑧 ↔ 𝑧 = 𝐵)
14	12, 13	bitri 183	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
15	11, 14	bitrdi 195	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
16	10, 15	syl5ib 153	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦 ∈ 𝐴 → 𝑧 = 𝐵))
17	8, 16	ralrimi 2528	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
18		eqeq1 2164	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
19	18	ralbidv 2457	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
20	17, 19	syl5ibrcom 156	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
21	5, 20	sylbid 149	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	exlimiv 1578	. . 3 ⊢ (∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23	1, 22	sylbi 120	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
24		euex 2036	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
25		rexm 3493	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 𝑦 ∈ 𝐴)
26	25	exlimiv 1578	. . 3 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 𝑦 ∈ 𝐴)
27		r19.2m 3480	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
28	27	ex 114	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
29	24, 26, 28	3syl 17	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
30	23, 29	impbid 128	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335  ∃wex 1472  ∃!weu 2006   ∈ wcel 2128  {cab 2143  ∀wral 2435  ∃wrex 2436  Vcvv 2712  {csn 3560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-sn 3566

This theorem is referenced by:  eusvobj1  5808