Theorem cbvmpov 7538

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5210, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpov.1	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
cbvmpov.2	⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)

Assertion

Ref	Expression
cbvmpov	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐷(𝑧, 𝑤)   𝐸(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem cbvmpov

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2847	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		eleq1w 2847	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
3	1, 2	bi2anan9 650	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
4		cbvmpov.1	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
5		cbvmpov.2	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
6	4, 5	sylan9eq 2818	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
7	6	eqeq2d 2774	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = 𝐷))
8	3, 7	anbi12d 644	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)))
9	8	cbvoprab12v 7533	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
10		df-mpo 7448	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
11		df-mpo 7448	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
12	9, 10, 11	3eqtr4i 2796	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {coprab 7444   ∈ cmpo 7445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-oprab 7447  df-mpo 7448

This theorem is used by:  fvproj  8178  seqomlem0  8487  dffi3  9450  cantnfsuc  9698  fin23lem33  10450  om2uzrdg  14126  uzrdgsuci  14130  sadcp1  16654  smupp1  16679  imasvscafn  17738  mgmnsgrpex  19159  sgrpnmndex  19160  sylow1  19847  sylow2b  19867  sylow3lem5  19875  sylow3  19877  efgmval  19956  efgtf  19966  funcrngcsetc  20922  funcrngcsetcALT  20923  funcringcsetc  20956  frlmphl  22117  pmatcollpw3lem  23131  mp2pm2mplem3  23156  txbas  23916  mpomulcn  25218  bcth  25680  opnmbl  25953  mbfimaopn  26007  mbfi1fseq  26072  om2noseqrdg  28720  noseqrdgsuc  28724  motplusg  29035  ttgval  29482  opsqrlem3  32774  elrgspnlem2  33874  splysubrg  34262  issply  34263  fedgmul  34333  mdetpmtr12  34527  madjusmdetlem4  34532  dya2iocival  34975  sxbrsigalem5  34990  sxbrsigalem6  34991  eulerpart  35084  sseqp1  35097  cvmliftlem15  36230  cvmlift2  36248  opnmbllem0  38766  mblfinlem1  38767  mblfinlem2  38768  sdc  38870  tendoplcbv  42024  dvhvaddcbv  42338  dvhvscacbv  42347  fsovcnvlem  45205  ntrneibex  45265  ioorrnopn  47520  hoidmvle  47815  ovnhoi  47818  hoimbl  47846  smflimlem6  47991  lmod1zr  49810  functhinclem4  50787  veronesematbasd  51260  veronesematrowd  51261  veroquadmodzerod  51264  veroquadnolindfd  51265  veroquaddetzerod  51266