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Theorem elixpsn 8919
Description: Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
elixpsn (𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦

Proof of Theorem elixpsn
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4634 . . . 4 (𝑧 = 𝐴 → {𝑧} = {𝐴})
21ixpeq1d 8891 . . 3 (𝑧 = 𝐴X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
32eleq2d 2820 . 2 (𝑧 = 𝐴 → (𝐹X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝐴}𝐵))
4 opeq1 4869 . . . . 5 (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
54sneqd 4636 . . . 4 (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
65eqeq2d 2744 . . 3 (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
76rexbidv 3179 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8 elex 3493 . . 3 (𝐹X𝑥 ∈ {𝑧}𝐵𝐹 ∈ V)
9 snex 5427 . . . . 5 {⟨𝑧, 𝑦⟩} ∈ V
10 eleq1 2822 . . . . 5 (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
119, 10mpbiri 258 . . . 4 (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
1211rexlimivw 3152 . . 3 (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13 eleq1 2822 . . . 4 (𝑤 = 𝐹 → (𝑤X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝑧}𝐵))
14 eqeq1 2737 . . . . 5 (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
1514rexbidv 3179 . . . 4 (𝑤 = 𝐹 → (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16 vex 3479 . . . . . 6 𝑤 ∈ V
1716elixp 8886 . . . . 5 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵))
18 vex 3479 . . . . . . 7 𝑧 ∈ V
19 fveq2 6881 . . . . . . . 8 (𝑥 = 𝑧 → (𝑤𝑥) = (𝑤𝑧))
2019eleq1d 2819 . . . . . . 7 (𝑥 = 𝑧 → ((𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2118, 20ralsn 4681 . . . . . 6 (∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2221anbi2i 624 . . . . 5 ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
23 simpl 484 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24 fveq2 6881 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑦) = (𝑤𝑧))
2524eleq1d 2819 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2618, 25ralsn 4681 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2726biimpri 227 . . . . . . . . . 10 ((𝑤𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
2827adantl 483 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
29 ffnfv 7105 . . . . . . . . 9 (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵))
3023, 28, 29sylanbrc 584 . . . . . . . 8 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
3118fsn2 7121 . . . . . . . 8 (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
3230, 31sylib 217 . . . . . . 7 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
33 opeq2 4870 . . . . . . . . 9 (𝑦 = (𝑤𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤𝑧)⟩)
3433sneqd 4636 . . . . . . . 8 (𝑦 = (𝑤𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤𝑧)⟩})
3534rspceeqv 3631 . . . . . . 7 (((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
3632, 35syl 17 . . . . . 6 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
37 vex 3479 . . . . . . . . . . 11 𝑦 ∈ V
3818, 37fvsn 7166 . . . . . . . . . 10 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
39 id 22 . . . . . . . . . 10 (𝑦𝐵𝑦𝐵)
4038, 39eqeltrid 2838 . . . . . . . . 9 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
4118, 37fnsn 6598 . . . . . . . . 9 {⟨𝑧, 𝑦⟩} Fn {𝑧}
4240, 41jctil 521 . . . . . . . 8 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
43 fneq1 6632 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
44 fveq1 6880 . . . . . . . . . 10 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
4544eleq1d 2819 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
4643, 45anbi12d 632 . . . . . . . 8 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
4742, 46syl5ibrcom 246 . . . . . . 7 (𝑦𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵)))
4847rexlimiv 3149 . . . . . 6 (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
4936, 48impbii 208 . . . . 5 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5017, 22, 493bitri 297 . . . 4 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5113, 15, 50vtoclbg 3558 . . 3 (𝐹 ∈ V → (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
528, 12, 51pm5.21nii 380 . 2 (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
533, 7, 52vtoclbg 3558 1 (𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  {csn 4624  cop 4630   Fn wfn 6530  wf 6531  cfv 6535  Xcixp 8879
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 5295  ax-nul 5302  ax-pr 5423
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-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ixp 8880
This theorem is referenced by:  ixpsnf1o  8920  hoidmv1le  45183
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