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Theorem elixpsn 8503
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 4579 . . . 4 (𝑧 = 𝐴 → {𝑧} = {𝐴})
21ixpeq1d 8475 . . 3 (𝑧 = 𝐴X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
32eleq2d 2900 . 2 (𝑧 = 𝐴 → (𝐹X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝐴}𝐵))
4 opeq1 4805 . . . . 5 (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
54sneqd 4581 . . . 4 (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
65eqeq2d 2834 . . 3 (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
76rexbidv 3299 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8 elex 3514 . . 3 (𝐹X𝑥 ∈ {𝑧}𝐵𝐹 ∈ V)
9 snex 5334 . . . . 5 {⟨𝑧, 𝑦⟩} ∈ V
10 eleq1 2902 . . . . 5 (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
119, 10mpbiri 260 . . . 4 (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
1211rexlimivw 3284 . . 3 (∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13 eleq1 2902 . . . 4 (𝑤 = 𝐹 → (𝑤X𝑥 ∈ {𝑧}𝐵𝐹X𝑥 ∈ {𝑧}𝐵))
14 eqeq1 2827 . . . . 5 (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
1514rexbidv 3299 . . . 4 (𝑤 = 𝐹 → (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16 vex 3499 . . . . . 6 𝑤 ∈ V
1716elixp 8470 . . . . 5 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵))
18 vex 3499 . . . . . . 7 𝑧 ∈ V
19 fveq2 6672 . . . . . . . 8 (𝑥 = 𝑧 → (𝑤𝑥) = (𝑤𝑧))
2019eleq1d 2899 . . . . . . 7 (𝑥 = 𝑧 → ((𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2118, 20ralsn 4621 . . . . . 6 (∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2221anbi2i 624 . . . . 5 ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
23 simpl 485 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24 fveq2 6672 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑦) = (𝑤𝑧))
2524eleq1d 2899 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵))
2618, 25ralsn 4621 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵 ↔ (𝑤𝑧) ∈ 𝐵)
2726biimpri 230 . . . . . . . . . 10 ((𝑤𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
2827adantl 484 . . . . . . . . 9 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵)
29 ffnfv 6884 . . . . . . . . 9 (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤𝑦) ∈ 𝐵))
3023, 28, 29sylanbrc 585 . . . . . . . 8 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
3118fsn2 6900 . . . . . . . 8 (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
3230, 31sylib 220 . . . . . . 7 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}))
33 opeq2 4806 . . . . . . . . 9 (𝑦 = (𝑤𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤𝑧)⟩)
3433sneqd 4581 . . . . . . . 8 (𝑦 = (𝑤𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤𝑧)⟩})
3534rspceeqv 3640 . . . . . . 7 (((𝑤𝑧) ∈ 𝐵𝑤 = {⟨𝑧, (𝑤𝑧)⟩}) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
3632, 35syl 17 . . . . . 6 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) → ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
37 vex 3499 . . . . . . . . . . 11 𝑦 ∈ V
3818, 37fvsn 6945 . . . . . . . . . 10 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
39 id 22 . . . . . . . . . 10 (𝑦𝐵𝑦𝐵)
4038, 39eqeltrid 2919 . . . . . . . . 9 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
4118, 37fnsn 6414 . . . . . . . . 9 {⟨𝑧, 𝑦⟩} Fn {𝑧}
4240, 41jctil 522 . . . . . . . 8 (𝑦𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
43 fneq1 6446 . . . . . . . . 9 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
44 fveq1 6671 . . . . . . . . . 10 (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
4544eleq1d 2899 . . . . . . . . 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 249 . . . . . . 7 (𝑦𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵)))
4847rexlimiv 3282 . . . . . 6 (∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵))
4936, 48impbii 211 . . . . 5 ((𝑤 Fn {𝑧} ∧ (𝑤𝑧) ∈ 𝐵) ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5017, 22, 493bitri 299 . . . 4 (𝑤X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
5113, 15, 50vtoclbg 3571 . . 3 (𝐹 ∈ V → (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
528, 12, 51pm5.21nii 382 . 2 (𝐹X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
533, 7, 52vtoclbg 3571 1 (𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3140  wrex 3141  Vcvv 3496  {csn 4569  cop 4575   Fn wfn 6352  wf 6353  cfv 6357  Xcixp 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ixp 8464
This theorem is referenced by:  ixpsnf1o  8504  hoidmv1le  42883
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