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Theorem indexa 36404
Description: If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
indexa ((𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐(𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑐   𝑥,𝐵,𝑦,𝑐   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑐)

Proof of Theorem indexa
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabexg 5324 . 2 (𝐵𝑀 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V)
2 ssrab2 4073 . . . 4 {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵
32a1i 11 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵)
4 nfv 1917 . . . . 5 𝑦 𝑥𝐴
5 nfre1 3281 . . . . 5 𝑦𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑
6 sbceq2a 3785 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ([𝑤 / 𝑥]𝜑𝜑))
76rspcev 3609 . . . . . . . . . . . . 13 ((𝑥𝐴𝜑) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
87ancoms 459 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
98anim1ci 616 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
109anasss 467 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1110ancoms 459 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
12 sbceq2a 3785 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
1312sbcbidv 3832 . . . . . . . . . . 11 (𝑧 = 𝑦 → ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑[𝑤 / 𝑥]𝜑))
1413rexbidv 3177 . . . . . . . . . 10 (𝑧 = 𝑦 → (∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1514elrab 3679 . . . . . . . . 9 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1611, 15sylibr 233 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → 𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑})
17 sbceq2a 3785 . . . . . . . . 9 (𝑣 = 𝑦 → ([𝑣 / 𝑦]𝜑𝜑))
1817rspcev 3609 . . . . . . . 8 ((𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∧ 𝜑) → ∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
1916, 18sylancom 588 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → ∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
20 nfcv 2902 . . . . . . . 8 𝑣{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
21 nfcv 2902 . . . . . . . . . 10 𝑦𝐴
22 nfcv 2902 . . . . . . . . . . 11 𝑦𝑤
23 nfsbc1v 3793 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
2422, 23nfsbcw 3795 . . . . . . . . . 10 𝑦[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
2521, 24nfrexw 3309 . . . . . . . . 9 𝑦𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
26 nfcv 2902 . . . . . . . . 9 𝑦𝐵
2725, 26nfrabw 3468 . . . . . . . 8 𝑦{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
28 nfsbc1v 3793 . . . . . . . 8 𝑦[𝑣 / 𝑦]𝜑
29 nfv 1917 . . . . . . . 8 𝑣𝜑
3020, 27, 28, 29, 17cbvrexfw 3301 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑 ↔ ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
3119, 30sylib 217 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
3231exp31 420 . . . . 5 (𝑥𝐴 → (𝑦𝐵 → (𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)))
334, 5, 32rexlimd 3262 . . . 4 (𝑥𝐴 → (∃𝑦𝐵 𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
3433ralimia 3079 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
35 nfsbc1v 3793 . . . . . . . . 9 𝑥[𝑤 / 𝑥]𝜑
36 nfv 1917 . . . . . . . . 9 𝑤𝜑
3735, 36, 6cbvrexw 3303 . . . . . . . 8 (∃𝑤𝐴 [𝑤 / 𝑥]𝜑 ↔ ∃𝑥𝐴 𝜑)
3814, 37bitrdi 286 . . . . . . 7 (𝑧 = 𝑦 → (∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 𝜑))
3938elrab 3679 . . . . . 6 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
4039simprbi 497 . . . . 5 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ∃𝑥𝐴 𝜑)
4140rgen 3062 . . . 4 𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑
4241a1i 11 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑)
433, 34, 423jca 1128 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → ({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑))
44 sseq1 4003 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (𝑐𝐵 ↔ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵))
45 nfcv 2902 . . . . . . . . 9 𝑥𝐴
46 nfsbc1v 3793 . . . . . . . . 9 𝑥[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
4745, 46nfrexw 3309 . . . . . . . 8 𝑥𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
48 nfcv 2902 . . . . . . . 8 𝑥𝐵
4947, 48nfrabw 3468 . . . . . . 7 𝑥{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
5049nfeq2 2919 . . . . . 6 𝑥 𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
51 nfcv 2902 . . . . . . 7 𝑦𝑐
5251, 27rexeqf 3349 . . . . . 6 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5350, 52ralbid 3269 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5451, 27raleqf 3348 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑))
5544, 53, 543anbi123d 1436 . . . 4 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ((𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ ({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑)))
5655spcegv 3584 . . 3 ({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V → (({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑) → ∃𝑐(𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5756imp 407 . 2 (({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V ∧ ({𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑)) → ∃𝑐(𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
581, 43, 57syl2an 596 1 ((𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐(𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  [wsbc 3773  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-in 3951  df-ss 3961
This theorem is referenced by: (None)
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