Theorem cbvmpov 7495

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5207, 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 2848	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		eleq1w 2848	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
3	1, 2	bi2anan9 649	. . . 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 2820	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
7	6	eqeq2d 2776	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = 𝐷))
8	3, 7	anbi12d 643	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)))
9	8	cbvoprab12v 7490	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
10		df-mpo 7405	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
11		df-mpo 7405	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
12	9, 10, 11	3eqtr4i 2798	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {coprab 7401   ∈ cmpo 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-oprab 7404  df-mpo 7405

This theorem is referenced by:  fvproj  8118  seqomlem0  8424  dffi3  9379  cantnfsuc  9627  fin23lem33  10317  om2uzrdg  13983  uzrdgsuci  13987  sadcp1  16503  smupp1  16528  imasvscafn  17581  mgmnsgrpex  18983  sgrpnmndex  18984  sylow1  19664  sylow2b  19684  sylow3lem5  19692  sylow3  19694  efgmval  19773  efgtf  19783  funcrngcsetc  20716  funcrngcsetcALT  20717  funcringcsetc  20750  frlmphl  21891  pmatcollpw3lem  22901  mp2pm2mplem3  22926  txbas  23685  mpomulcn  24987  bcth  25449  opnmbl  25722  mbfimaopn  25776  mbfi1fseq  25841  om2noseqrdg  28455  noseqrdgsuc  28459  motplusg  28769  ttgval  29133  opsqrlem3  32403  elrgspnlem2  33476  splysubrg  33867  issply  33868  fedgmul  33938  mdetpmtr12  34132  madjusmdetlem4  34137  dya2iocival  34580  sxbrsigalem5  34595  sxbrsigalem6  34596  eulerpart  34689  sseqp1  34702  cvmliftlem15  35661  cvmlift2  35679  opnmbllem0  38167  mblfinlem1  38168  mblfinlem2  38169  sdc  38255  tendoplcbv  41411  dvhvaddcbv  41725  dvhvscacbv  41734  fsovcnvlem  44601  ntrneibex  44661  ioorrnopn  46877  hoidmvle  47172  ovnhoi  47175  hoimbl  47203  smflimlem6  47348  lmod1zr  49124  functhinclem4  50076