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Theorem indexa 37720
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 5343 . 2 (𝐵𝑀 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V)
2 ssrab2 4090 . . . 4 {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵
32a1i 11 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵)
4 nfv 1912 . . . . 5 𝑦 𝑥𝐴
5 nfre1 3283 . . . . 5 𝑦𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑
6 sbceq2a 3803 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ([𝑤 / 𝑥]𝜑𝜑))
76rspcev 3622 . . . . . . . . . . . . 13 ((𝑥𝐴𝜑) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
87ancoms 458 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∃𝑤𝐴 [𝑤 / 𝑥]𝜑)
98anim1ci 616 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
109anasss 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1110ancoms 458 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
12 sbceq2a 3803 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
1312sbcbidv 3851 . . . . . . . . . . 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 3695 . . . . . . . . 9 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦𝐵 ∧ ∃𝑤𝐴 [𝑤 / 𝑥]𝜑))
1611, 15sylibr 234 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → 𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑})
17 sbceq2a 3803 . . . . . . . . 9 (𝑣 = 𝑦 → ([𝑣 / 𝑦]𝜑𝜑))
1817rspcev 3622 . . . . . . . 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 2903 . . . . . . . 8 𝑣{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
21 nfcv 2903 . . . . . . . . . 10 𝑦𝐴
22 nfcv 2903 . . . . . . . . . . 11 𝑦𝑤
23 nfsbc1v 3811 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
2422, 23nfsbcw 3813 . . . . . . . . . 10 𝑦[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
2521, 24nfrexw 3311 . . . . . . . . 9 𝑦𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
26 nfcv 2903 . . . . . . . . 9 𝑦𝐵
2725, 26nfrabw 3473 . . . . . . . 8 𝑦{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
28 nfsbc1v 3811 . . . . . . . 8 𝑦[𝑣 / 𝑦]𝜑
29 nfv 1912 . . . . . . . 8 𝑣𝜑
3020, 27, 28, 29, 17cbvrexfw 3303 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑 ↔ ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
3119, 30sylib 218 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
3231exp31 419 . . . . 5 (𝑥𝐴 → (𝑦𝐵 → (𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)))
334, 5, 32rexlimd 3264 . . . 4 (𝑥𝐴 → (∃𝑦𝐵 𝜑 → ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
3433ralimia 3078 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
35 nfsbc1v 3811 . . . . . . . . 9 𝑥[𝑤 / 𝑥]𝜑
36 nfv 1912 . . . . . . . . 9 𝑤𝜑
3735, 36, 6cbvrexw 3305 . . . . . . . 8 (∃𝑤𝐴 [𝑤 / 𝑥]𝜑 ↔ ∃𝑥𝐴 𝜑)
3814, 37bitrdi 287 . . . . . . 7 (𝑧 = 𝑦 → (∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 𝜑))
3938elrab 3695 . . . . . 6 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
4039simprbi 496 . . . . 5 (𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ∃𝑥𝐴 𝜑)
4140rgen 3061 . . . 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 1127 . 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 4021 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (𝑐𝐵 ↔ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵))
45 nfcv 2903 . . . . . . . . 9 𝑥𝐴
46 nfsbc1v 3811 . . . . . . . . 9 𝑥[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
4745, 46nfrexw 3311 . . . . . . . 8 𝑥𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
48 nfcv 2903 . . . . . . . 8 𝑥𝐵
4947, 48nfrabw 3473 . . . . . . 7 𝑥{𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
5049nfeq2 2921 . . . . . 6 𝑥 𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
51 nfcv 2903 . . . . . . 7 𝑦𝑐
5251, 27rexeqf 3352 . . . . . 6 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5350, 52ralbid 3271 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
5451, 27raleqf 3351 . . . . 5 (𝑐 = {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ {𝑧𝐵 ∣ ∃𝑤𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥𝐴 𝜑))
5544, 53, 543anbi123d 1435 . . . 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 3597 . . 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 406 . 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 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  [wsbc 3791  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-in 3970  df-ss 3980  df-pw 4607
This theorem is referenced by: (None)
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