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Theorem funopdmsn 7143
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 6566 . . . 4 (Fun 𝐺 ↔ Fun ⟨𝑋, 𝑌⟩)
3 funopdmsn.x . . . . . 6 𝑋𝑉
43elexi 3494 . . . . 5 𝑋 ∈ V
5 funopdmsn.y . . . . . 6 𝑌𝑊
65elexi 3494 . . . . 5 𝑌 ∈ V
74, 6funop 7142 . . . 4 (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
82, 7bitri 275 . . 3 (Fun 𝐺 ↔ ∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}))
91eqcomi 2742 . . . . . . 7 𝑋, 𝑌⟩ = 𝐺
109eqeq1i 2738 . . . . . 6 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} ↔ 𝐺 = {⟨𝑥, 𝑥⟩})
11 dmeq 5901 . . . . . . . 8 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = dom {⟨𝑥, 𝑥⟩})
12 vex 3479 . . . . . . . . 9 𝑥 ∈ V
1312dmsnop 6212 . . . . . . . 8 dom {⟨𝑥, 𝑥⟩} = {𝑥}
1411, 13eqtrdi 2789 . . . . . . 7 (𝐺 = {⟨𝑥, 𝑥⟩} → dom 𝐺 = {𝑥})
15 eleq2 2823 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐴 ∈ dom 𝐺𝐴 ∈ {𝑥}))
16 eleq2 2823 . . . . . . . . 9 (dom 𝐺 = {𝑥} → (𝐵 ∈ dom 𝐺𝐵 ∈ {𝑥}))
1715, 16anbi12d 632 . . . . . . . 8 (dom 𝐺 = {𝑥} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) ↔ (𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥})))
18 elsni 4644 . . . . . . . . 9 (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
19 elsni 4644 . . . . . . . . 9 (𝐵 ∈ {𝑥} → 𝐵 = 𝑥)
20 eqtr3 2759 . . . . . . . . 9 ((𝐴 = 𝑥𝐵 = 𝑥) → 𝐴 = 𝐵)
2118, 19, 20syl2an 597 . . . . . . . 8 ((𝐴 ∈ {𝑥} ∧ 𝐵 ∈ {𝑥}) → 𝐴 = 𝐵)
2217, 21syl6bi 253 . . . . . . 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 216 . . . . 5 (⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩} → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2524adantl 483 . . . 4 ((𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
2625exlimiv 1934 . . 3 (∃𝑥(𝑋 = {𝑥} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑥, 𝑥⟩}) → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
278, 26sylbi 216 . 2 (Fun 𝐺 → ((𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵))
28273impib 1117 1 ((Fun 𝐺𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {csn 4627  cop 4633  dom cdm 5675  Fun wfun 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548
This theorem is referenced by:  fundmge2nop0  14449
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