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Theorem elixpsn 7989
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 4220 . . . 4 (𝑧 = 𝐴 → {𝑧} = {𝐴})
21ixpeq1d 7962 . . 3 (𝑧 = 𝐴X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
32eleq2d 2716 . 2 (𝑧 = 𝐴 → (𝐹X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝐴}𝐵))
4 opeq1 4433 . . . . 5 (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
54sneqd 4222 . . . 4 (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
65eqeq2d 2661 . . 3 (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
76rexbidv 3081 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8 elex 3243 . . 3 (𝐹X𝑥 ∈ {𝑧}𝐵𝐹 ∈ V)
9 snex 4938 . . . . 5 {⟨𝑧, 𝑦⟩} ∈ V
10 eleq1 2718 . . . . 5 (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
119, 10mpbiri 248 . . . 4 (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
1211rexlimivw 3058 . . 3 (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13 eleq1 2718 . . . 4 (𝑤 = 𝐹 → (𝑤X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝑧}𝐵))
14 eqeq1 2655 . . . . 5 (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
1514rexbidv 3081 . . . 4 (𝑤 = 𝐹 → (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16 vex 3234 . . . . . 6 𝑤 ∈ V
1716elixp 7957 . . . . 5 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵))
18 vex 3234 . . . . . . 7 𝑧 ∈ V
19 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑧 → (𝑤𝑥) = (𝑤𝑧))
2019eleq1d 2715 . . . . . . 7 (𝑥 = 𝑧 → ((𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2118, 20ralsn 4254 . . . . . 6 (∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2221anbi2i 730 . . . . 5 ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
23 simpl 472 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑦) = (𝑤𝑧))
2524eleq1d 2715 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2618, 25ralsn 4254 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2726biimpri 218 . . . . . . . . . 10 ((𝑤𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
2827adantl 481 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
29 ffnfv 6428 . . . . . . . . 9 (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵))
3023, 28, 29sylanbrc 699 . . . . . . . 8 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
3118fsn2 6443 . . . . . . . 8 (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
3230, 31sylib 208 . . . . . . 7 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
33 opeq2 4434 . . . . . . . . . 10 (𝑦 = (𝑤𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤𝑧)⟩)
3433sneqd 4222 . . . . . . . . 9 (𝑦 = (𝑤𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤𝑧)⟩})
3534eqeq2d 2661 . . . . . . . 8 (𝑦 = (𝑤𝑧) → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
3635rspcev 3340 . . . . . . 7 (((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
3732, 36syl 17 . . . . . 6 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
38 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
3918, 38fvsn 6487 . . . . . . . . . 10 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
40 id 22 . . . . . . . . . 10 (𝑦𝐵𝑦𝐵)
4139, 40syl5eqel 2734 . . . . . . . . 9 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
4218, 38fnsn 5984 . . . . . . . . 9 {⟨𝑧, 𝑦⟩} Fn {𝑧}
4341, 42jctil 559 . . . . . . . 8 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
44 fneq1 6017 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
45 fveq1 6228 . . . . . . . . . 10 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
4645eleq1d 2715 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
4744, 46anbi12d 747 . . . . . . . 8 (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
4843, 47syl5ibrcom 237 . . . . . . 7 (𝑦𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵)))
4948rexlimiv 3056 . . . . . 6 (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
5037, 49impbii 199 . . . . 5 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5117, 22, 503bitri 286 . . . 4 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5213, 15, 51vtoclbg 3298 . . 3 (𝐹 ∈ V → (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
538, 12, 52pm5.21nii 367 . 2 (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
543, 7, 53vtoclbg 3298 1 (𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  {csn 4210  cop 4216   Fn wfn 5921  wf 5922  cfv 5926  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ixp 7951
This theorem is referenced by:  ixpsnf1o  7990  hoidmv1le  41129
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