Theorem funopdmsn 7144

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 6563	. . . 4 ⊢ (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3		funopdmsn.x	. . . . . 6 ⊢ 𝑋 ∈ 𝑉
4	3	elexi 3488	. . . . 5 ⊢ 𝑋 ∈ V
5		funopdmsn.y	. . . . . 6 ⊢ 𝑌 ∈ 𝑊
6	5	elexi 3488	. . . . 5 ⊢ 𝑌 ∈ V
7	4, 6	funop 7143	. . . 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 2735	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ = 𝐺
10	9	eqeq1i 2731	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11		dmeq 5897	. . . . . . . 8 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12		vex 3472	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	dmsnop 6209	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
14	11, 13	eqtrdi 2782	. . . . . . 7 ⊢ (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15		eleq2 2816	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺 ↔ 𝐴 ∈ {𝑥}))
16		eleq2 2816	. . . . . . . . 9 ⊢ (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺 ↔ 𝐵 ∈ {𝑥}))
17	15, 16	anbi12d 630	. . . . . . . 8 ⊢ (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18		elsni 4640	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19		elsni 4640	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20		eqtr3 2752	. . . . . . . . 9 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑥) → 𝐴 = 𝐵)
21	18, 19, 20	syl2an 595	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
22	17, 21	biimtrdi 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 481	. . . 4 ⊢ ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
26	25	exlimiv 1925	. . 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 1113	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {csn 4623  ⟨cop 4629  dom cdm 5669  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  fundmge2nop0  14459