Theorem funopdmsn 7095

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 6511	. . . 4 ⊢ (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3		funopdmsn.x	. . . . . 6 ⊢ 𝑋 ∈ 𝑉
4	3	elexi 3453	. . . . 5 ⊢ 𝑋 ∈ V
5		funopdmsn.y	. . . . . 6 ⊢ 𝑌 ∈ 𝑊
6	5	elexi 3453	. . . . 5 ⊢ 𝑌 ∈ V
7	4, 6	funop 7094	. . . 4 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
8	2, 7	bitri 275	. . 3 ⊢ (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
9	1	eqcomi 2746	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ = 𝐺
10	9	eqeq1i 2742	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11		dmeq 5850	. . . . . . . 8 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12		vex 3434	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	dmsnop 6172	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
14	11, 13	eqtrdi 2788	. . . . . . 7 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15		eleq2 2826	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺 ↔ 𝐴 ∈ {𝑥}))
16		eleq2 2826	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺 ↔ 𝐵 ∈ {𝑥}))
17	15, 16	anbi12d 633	. . . . . . . 8 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18		elsni 4585	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19		elsni 4585	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20		eqtr3 2759	. . . . . . . . 9 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑥) → 𝐴 = 𝐵)
21	18, 19, 20	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
22	17, 21	biimtrdi 253	. . . . . . 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 217	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
25	24	adantl 481	. . . 4 ⊢ ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
26	25	exlimiv 1932	. . 3 ⊢ (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
27	8, 26	sylbi 217	. 2 ⊢ (Fun 𝐺 → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28	27	3impib 1117	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {csn 4568  ⟨cop 4574  dom cdm 5622  Fun wfun 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498

This theorem is referenced by:  fundmge2nop0  14453