Theorem funopdmsn 5777

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 5301	. . . 4 ⊢ (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3		funopdmsn.x	. . . . . 6 ⊢ 𝑋 ∈ 𝑉
4	3	elexi 2786	. . . . 5 ⊢ 𝑋 ∈ V
5		funopdmsn.y	. . . . . 6 ⊢ 𝑌 ∈ 𝑊
6	5	elexi 2786	. . . . 5 ⊢ 𝑌 ∈ V
7	4, 6	funop 5776	. . . 4 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
8	2, 7	bitri 184	. . 3 ⊢ (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
9	1	eqcomi 2210	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ = 𝐺
10	9	eqeq1i 2214	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11		dmeq 4887	. . . . . . . 8 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12		vex 2776	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	dmsnop 5165	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
14	11, 13	eqtrdi 2255	. . . . . . 7 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15		eleq2 2270	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺 ↔ 𝐴 ∈ {𝑥}))
16		eleq2 2270	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺 ↔ 𝐵 ∈ {𝑥}))
17	15, 16	anbi12d 473	. . . . . . . 8 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18		elsni 3656	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19		elsni 3656	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20		eqtr3 2226	. . . . . . . . 9 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑥) → 𝐴 = 𝐵)
21	18, 19, 20	syl2an 289	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
22	17, 21	biimtrdi 163	. . . . . . 7 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
23	14, 22	syl 14	. . . . . 6 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
24	10, 23	sylbi 121	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
25	24	adantl 277	. . . 4 ⊢ ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
26	25	exlimiv 1622	. . 3 ⊢ (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
27	8, 26	sylbi 121	. 2 ⊢ (Fun 𝐺 → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28	27	3impib 1204	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {csn 3638  ⟨cop 3641  dom cdm 4683  Fun wfun 5274

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-fun 5282

This theorem is referenced by:  fundm2domnop0  11012