Theorem eusvobj2 7403

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 4728	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧})
2		eleq2 2820	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3		abid 2711	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		velsn 4643	. . . . . 6 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
5	2, 3, 4	3bitr3g 312	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ 𝑥 = 𝑧))
6		nfre1 3280	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵
7	6	nfab 2907	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵}
8	7	nfeq1 2916	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧}
9		eusvobj1.1	. . . . . . . . 9 ⊢ 𝐵 ∈ V
10	9	elabrex 7243	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵})
11		eleq2 2820	. . . . . . . . 9 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
12	9	elsn 4642	. . . . . . . . . 10 ⊢ (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13		eqcom 2737	. . . . . . . . . 10 ⊢ (𝐵 = 𝑧 ↔ 𝑧 = 𝐵)
14	12, 13	bitri 274	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
15	11, 14	bitrdi 286	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
16	10, 15	imbitrid 243	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦 ∈ 𝐴 → 𝑧 = 𝐵))
17	8, 16	ralrimi 3252	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
18		eqeq1 2734	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
19	18	ralbidv 3175	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
20	17, 19	syl5ibrcom 246	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
21	5, 20	sylbid 239	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	exlimiv 1931	. . 3 ⊢ (∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23	1, 22	sylbi 216	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
24		euex 2569	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
25		rexn0 4509	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
26	25	exlimiv 1931	. . 3 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
27		r19.2z 4493	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
28	27	ex 411	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
29	24, 26, 28	3syl 18	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
30	23, 29	impbid 211	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃!weu 2560  {cab 2707   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3472  ∅c0 4321  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-nul 4322  df-sn 4628

This theorem is referenced by:  eusvobj1  7404