Theorem funopdmsn 7022

Description: The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
funopdmsn.g	⊢ 𝐺 = ⟨𝑋, 𝑌⟩
funopdmsn.x	⊢ 𝑋 ∈ 𝑉
funopdmsn.y	⊢ 𝑌 ∈ 𝑊

Assertion

Ref	Expression
funopdmsn	⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Proof of Theorem funopdmsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopdmsn.g	. . . . 5 ⊢ 𝐺 = ⟨𝑋, 𝑌⟩
2	1	funeqi 6455	. . . 4 ⊢ (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3		funopdmsn.x	. . . . . 6 ⊢ 𝑋 ∈ 𝑉
4	3	elexi 3451	. . . . 5 ⊢ 𝑋 ∈ V
5		funopdmsn.y	. . . . . 6 ⊢ 𝑌 ∈ 𝑊
6	5	elexi 3451	. . . . 5 ⊢ 𝑌 ∈ V
7	4, 6	funop 7021	. . . 4 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
8	2, 7	bitri 274	. . 3 ⊢ (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
9	1	eqcomi 2747	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ = 𝐺
10	9	eqeq1i 2743	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11		dmeq 5812	. . . . . . . 8 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	dmsnop 6119	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
14	11, 13	eqtrdi 2794	. . . . . . 7 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15		eleq2 2827	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺 ↔ 𝐴 ∈ {𝑥}))
16		eleq2 2827	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺 ↔ 𝐵 ∈ {𝑥}))
17	15, 16	anbi12d 631	. . . . . . . 8 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18		elsni 4578	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19		elsni 4578	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20		eqtr3 2764	. . . . . . . . 9 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑥) → 𝐴 = 𝐵)
21	18, 19, 20	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
22	17, 21	syl6bi 252	. . . . . . 7 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
23	14, 22	syl 17	. . . . . 6 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
24	10, 23	sylbi 216	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
25	24	adantl 482	. . . 4 ⊢ ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
26	25	exlimiv 1933	. . 3 ⊢ (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
27	8, 26	sylbi 216	. 2 ⊢ (Fun 𝐺 → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28	27	3impib 1115	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {csn 4561  ⟨cop 4567  dom cdm 5589  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  fundmge2nop0  14206