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Theorem funopdmsn 6580
Description: The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.)
Hypotheses
Ref Expression
funopdmsn.g 𝐺 = ⟨𝑋, 𝑌
funopdmsn.x 𝑋𝑉
funopdmsn.y 𝑌𝑊
Assertion
Ref Expression
funopdmsn ((Fun 𝐺𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Proof of Theorem funopdmsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funopdmsn.g . . . . 5 𝐺 = ⟨𝑋, 𝑌
21funeqi 6071 . . . 4 (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3 funopdmsn.x . . . . . 6 𝑋𝑉
43elexi 3354 . . . . 5 𝑋 ∈ V
5 funopdmsn.y . . . . . 6 𝑌𝑊
65elexi 3354 . . . . 5 𝑌 ∈ V
74, 6funop 6579 . . . 4 (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
82, 7bitri 264 . . 3 (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
91eqcomi 2770 . . . . . . 7 𝑋, 𝑌⟩ = 𝐺
109eqeq1i 2766 . . . . . 6 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11 dmeq 5480 . . . . . . . 8 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12 vex 3344 . . . . . . . . 9 𝑥 ∈ V
1312dmsnop 5769 . . . . . . . 8 dom {⟨𝑥, 𝑥⟩} = {𝑥}
1411, 13syl6eq 2811 . . . . . . 7 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15 eleq2 2829 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺𝐴 ∈ {𝑥}))
16 eleq2 2829 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺𝐵 ∈ {𝑥}))
1715, 16anbi12d 749 . . . . . . . 8 (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18 elsni 4339 . . . . . . . . 9 (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19 elsni 4339 . . . . . . . . 9 (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20 eqtr3 2782 . . . . . . . . 9 ((𝐴 = 𝑥𝐵 = 𝑥) → 𝐴 = 𝐵)
2118, 19, 20syl2an 495 . . . . . . . 8 ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
2217, 21syl6bi 243 . . . . . . 7 (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2314, 22syl 17 . . . . . 6 (𝐺 = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2410, 23sylbi 207 . . . . 5 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2524adantl 473 . . . 4 ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2625exlimiv 2008 . . 3 (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
278, 26sylbi 207 . 2 (Fun 𝐺 → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28273impib 1109 1 ((Fun 𝐺𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2140  {csn 4322  cop 4328  dom cdm 5267  Fun wfun 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058
This theorem is referenced by:  fundmge2nop0  13487
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