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Theorem indexa 36242
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 5292 . 2 (𝐵𝑀 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V)
2 ssrab2 4041 . . . 4 {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵
32a1i 11 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵)
4 nfv 1918 . . . . 5 𝑦 𝑥𝐴
5 nfre1 3267 . . . . 5 𝑦𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑
6 sbceq2a 3755 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ([𝑤 / 𝑥]𝜑𝜑))
76rspcev 3583 . . . . . . . . . . . . 13 ((𝑥𝐴𝜑) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
87ancoms 460 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
98anim1ci 617 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
109anasss 468 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1110ancoms 460 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
12 sbceq2a 3755 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
1312sbcbidv 3802 . . . . . . . . . . 11 (𝑧 = 𝑦 → ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑[𝑤 / 𝑥]𝜑))
1413rexbidv 3172 . . . . . . . . . 10 (𝑧 = 𝑦 → (∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1514elrab 3649 . . . . . . . . 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 3755 . . . . . . . . 9 (𝑣 = 𝑦 → ([𝑣 / 𝑦]𝜑𝜑))
1817rspcev 3583 . . . . . . . 8 ((𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∧ 𝜑) → ∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
1916, 18sylancom 589 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → ∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
20 nfcv 2904 . . . . . . . 8 𝑣{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
21 nfcv 2904 . . . . . . . . . 10 𝑦𝐴
22 nfcv 2904 . . . . . . . . . . 11 𝑦𝑤
23 nfsbc1v 3763 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
2422, 23nfsbcw 3765 . . . . . . . . . 10 𝑦[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
2521, 24nfrexw 3295 . . . . . . . . 9 𝑦𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
26 nfcv 2904 . . . . . . . . 9 𝑦𝐵
2725, 26nfrabw 3442 . . . . . . . 8 𝑦{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
28 nfsbc1v 3763 . . . . . . . 8 𝑦[𝑣 / 𝑦]𝜑
29 nfv 1918 . . . . . . . 8 𝑣𝜑
3020, 27, 28, 29, 17cbvrexfw 3287 . . . . . . 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 421 . . . . 5 (𝑥𝐴 → (𝑦𝐵 → (𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)))
334, 5, 32rexlimd 3248 . . . 4 (𝑥𝐴 → (∃𝑦𝐵 𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
3433ralimia 3080 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
35 nfsbc1v 3763 . . . . . . . . 9 𝑥[𝑤 / 𝑥]𝜑
36 nfv 1918 . . . . . . . . 9 𝑤𝜑
3735, 36, 6cbvrexw 3289 . . . . . . . 8 (∃𝑤𝐴 [𝑤 / 𝑥]𝜑 ↔ ∃𝑥𝐴 𝜑)
3814, 37bitrdi 287 . . . . . . 7 (𝑧 = 𝑦 → (∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 𝜑))
3938elrab 3649 . . . . . 6 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
4039simprbi 498 . . . . 5 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ∃𝑥𝐴 𝜑)
4140rgen 3063 . . . 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 1129 . 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 3973 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (𝑐𝐵 ↔ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵))
45 nfcv 2904 . . . . . . . . 9 𝑥𝐴
46 nfsbc1v 3763 . . . . . . . . 9 𝑥[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
4745, 46nfrexw 3295 . . . . . . . 8 𝑥𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
48 nfcv 2904 . . . . . . . 8 𝑥𝐵
4947, 48nfrabw 3442 . . . . . . 7 𝑥{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
5049nfeq2 2921 . . . . . 6 𝑥 𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
51 nfcv 2904 . . . . . . 7 𝑦𝑐
5251, 27rexeqf 3328 . . . . . 6 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5350, 52ralbid 3255 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5451, 27raleqf 3327 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑))
5544, 53, 543anbi123d 1437 . . . 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 3558 . . 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 408 . 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 597 1 ((𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐(𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  [wsbc 3743  wss 3914
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 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-in 3921  df-ss 3931
This theorem is referenced by: (None)
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