Theorem funopdmsn 7151

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 6558	. . . 4 ⊢ (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3		funopdmsn.x	. . . . . 6 ⊢ 𝑋 ∈ 𝑉
4	3	elexi 3475	. . . . 5 ⊢ 𝑋 ∈ V
5		funopdmsn.y	. . . . . 6 ⊢ 𝑌 ∈ 𝑊
6	5	elexi 3475	. . . . 5 ⊢ 𝑌 ∈ V
7	4, 6	funop 7150	. . . 4 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
8	2, 7	bitri 278	. . 3 ⊢ (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
9	1	eqcomi 2771	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ = 𝐺
10	9	eqeq1i 2767	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11		dmeq 5891	. . . . . . . 8 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	dmsnop 6216	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
14	11, 13	eqtrdi 2813	. . . . . . 7 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15		eleq2 2851	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺 ↔ 𝐴 ∈ {𝑥}))
16		eleq2 2851	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺 ↔ 𝐵 ∈ {𝑥}))
17	15, 16	anbi12d 644	. . . . . . . 8 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18		elsni 4604	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19		elsni 4604	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20		eqtr3 2784	. . . . . . . . 9 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑥) → 𝐴 = 𝐵)
21	18, 19, 20	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
22	17, 21	biimtrdi 256	. . . . . . 7 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
23	14, 22	syl 18	. . . . . 6 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
24	10, 23	sylbi 220	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
25	24	adantl 487	. . . 4 ⊢ ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
26	25	exlimiv 1963	. . 3 ⊢ (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
27	8, 26	sylbi 220	. 2 ⊢ (Fun 𝐺 → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28	27	3impib 1134	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {csn 4587  ⟨cop 4593  dom cdm 5659  Fun wfun 6531

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  fundmge2nop0  14571