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Theorem elixpsn 8489
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 4567 . . . 4 (𝑧 = 𝐴 → {𝑧} = {𝐴})
21ixpeq1d 8461 . . 3 (𝑧 = 𝐴X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
32eleq2d 2895 . 2 (𝑧 = 𝐴 → (𝐹X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝐴}𝐵))
4 opeq1 4795 . . . . 5 (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
54sneqd 4569 . . . 4 (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
65eqeq2d 2829 . . 3 (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
76rexbidv 3294 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8 elex 3510 . . 3 (𝐹X𝑥 ∈ {𝑧}𝐵𝐹 ∈ V)
9 snex 5322 . . . . 5 {⟨𝑧, 𝑦⟩} ∈ V
10 eleq1 2897 . . . . 5 (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
119, 10mpbiri 259 . . . 4 (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
1211rexlimivw 3279 . . 3 (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13 eleq1 2897 . . . 4 (𝑤 = 𝐹 → (𝑤X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝑧}𝐵))
14 eqeq1 2822 . . . . 5 (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
1514rexbidv 3294 . . . 4 (𝑤 = 𝐹 → (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16 vex 3495 . . . . . 6 𝑤 ∈ V
1716elixp 8456 . . . . 5 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵))
18 vex 3495 . . . . . . 7 𝑧 ∈ V
19 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑧 → (𝑤𝑥) = (𝑤𝑧))
2019eleq1d 2894 . . . . . . 7 (𝑥 = 𝑧 → ((𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2118, 20ralsn 4611 . . . . . 6 (∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2221anbi2i 622 . . . . 5 ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
23 simpl 483 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24 fveq2 6663 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑦) = (𝑤𝑧))
2524eleq1d 2894 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2618, 25ralsn 4611 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2726biimpri 229 . . . . . . . . . 10 ((𝑤𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
2827adantl 482 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
29 ffnfv 6874 . . . . . . . . 9 (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵))
3023, 28, 29sylanbrc 583 . . . . . . . 8 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
3118fsn2 6890 . . . . . . . 8 (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
3230, 31sylib 219 . . . . . . 7 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
33 opeq2 4796 . . . . . . . . 9 (𝑦 = (𝑤𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤𝑧)⟩)
3433sneqd 4569 . . . . . . . 8 (𝑦 = (𝑤𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤𝑧)⟩})
3534rspceeqv 3635 . . . . . . 7 (((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
3632, 35syl 17 . . . . . 6 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
37 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
3818, 37fvsn 6935 . . . . . . . . . 10 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
39 id 22 . . . . . . . . . 10 (𝑦𝐵𝑦𝐵)
4038, 39eqeltrid 2914 . . . . . . . . 9 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
4118, 37fnsn 6405 . . . . . . . . 9 {⟨𝑧, 𝑦⟩} Fn {𝑧}
4240, 41jctil 520 . . . . . . . 8 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
43 fneq1 6437 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
44 fveq1 6662 . . . . . . . . . 10 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
4544eleq1d 2894 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
4643, 45anbi12d 630 . . . . . . . 8 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
4742, 46syl5ibrcom 248 . . . . . . 7 (𝑦𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵)))
4847rexlimiv 3277 . . . . . 6 (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
4936, 48impbii 210 . . . . 5 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5017, 22, 493bitri 298 . . . 4 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5113, 15, 50vtoclbg 3566 . . 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 3566 1 (𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  {csn 4557  cop 4563   Fn wfn 6343  wf 6344  cfv 6348  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ixp 8450
This theorem is referenced by:  ixpsnf1o  8490  hoidmv1le  42753
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