Theorem eusvobj2 7400

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 4685	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧})
2		eleq2 2849	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3		abid 2742	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		velsn 4599	. . . . . 6 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
5	2, 3, 4	3bitr3g 316	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ 𝑥 = 𝑧))
6		nfre1 3287	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵
7	6	nfab 2928	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵}
8	7	nfeq1 2937	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧}
9		eusvobj1.1	. . . . . . . . 9 ⊢ 𝐵 ∈ V
10	9	elabrex 7234	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵})
11		eleq2 2849	. . . . . . . . 9 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
12	9	elsn 4598	. . . . . . . . . 10 ⊢ (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13		eqcom 2767	. . . . . . . . . 10 ⊢ (𝐵 = 𝑧 ↔ 𝑧 = 𝐵)
14	12, 13	bitri 278	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
15	11, 14	bitrdi 290	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
16	10, 15	imbitrid 247	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦 ∈ 𝐴 → 𝑧 = 𝐵))
17	8, 16	ralrimi 3260	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
18		eqeq1 2764	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
19	18	ralbidv 3185	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
20	17, 19	syl5ibrcom 250	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
21	5, 20	sylbid 243	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	exlimiv 1963	. . 3 ⊢ (∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23	1, 22	sylbi 220	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
24		euex 2602	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
25		rexn0 4451	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
26	25	exlimiv 1963	. . 3 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
27		r19.2z 4454	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
28	27	ex 418	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
29	24, 26, 28	3syl 19	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
30	23, 29	impbid 215	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2593  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4278  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-nul 4279  df-sn 4584

This theorem is used by:  eusvobj1  7401