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Theorem funopdmsn 6903
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 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 6364 . . . 4 (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3 funopdmsn.x . . . . . 6 𝑋𝑉
43elexi 3499 . . . . 5 𝑋 ∈ V
5 funopdmsn.y . . . . . 6 𝑌𝑊
65elexi 3499 . . . . 5 𝑌 ∈ V
74, 6funop 6902 . . . 4 (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
82, 7bitri 278 . . 3 (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
91eqcomi 2833 . . . . . . 7 𝑋, 𝑌⟩ = 𝐺
109eqeq1i 2829 . . . . . 6 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11 dmeq 5759 . . . . . . . 8 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12 vex 3483 . . . . . . . . 9 𝑥 ∈ V
1312dmsnop 6060 . . . . . . . 8 dom {⟨𝑥, 𝑥⟩} = {𝑥}
1411, 13syl6eq 2875 . . . . . . 7 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15 eleq2 2904 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺𝐴 ∈ {𝑥}))
16 eleq2 2904 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺𝐵 ∈ {𝑥}))
1715, 16anbi12d 633 . . . . . . . 8 (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18 elsni 4567 . . . . . . . . 9 (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19 elsni 4567 . . . . . . . . 9 (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20 eqtr3 2846 . . . . . . . . 9 ((𝐴 = 𝑥𝐵 = 𝑥) → 𝐴 = 𝐵)
2118, 19, 20syl2an 598 . . . . . . . 8 ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
2217, 21syl6bi 256 . . . . . . 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 220 . . . . 5 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2524adantl 485 . . . 4 ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2625exlimiv 1932 . . 3 (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
278, 26sylbi 220 . 2 (Fun 𝐺 → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28273impib 1113 1 ((Fun 𝐺𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {csn 4550  cop 4556  dom cdm 5542  Fun wfun 6337
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351
This theorem is referenced by:  fundmge2nop0  13855
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