Theorem cbvmpov 7518

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5218, 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 2849	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		eleq1w 2849	. . . . 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 2821	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
7	6	eqeq2d 2777	. . . 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 7513	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
10		df-mpo 7428	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
11		df-mpo 7428	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
12	9, 10, 11	3eqtr4i 2799	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {coprab 7424   ∈ cmpo 7425

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-oprab 7427  df-mpo 7428

This theorem is used by:  fvproj  8139  seqomlem0  8445  dffi3  9401  cantnfsuc  9649  fin23lem33  10347  om2uzrdg  14012  uzrdgsuci  14016  sadcp1  16538  smupp1  16563  imasvscafn  17616  mgmnsgrpex  19024  sgrpnmndex  19025  sylow1  19704  sylow2b  19724  sylow3lem5  19732  sylow3  19734  efgmval  19813  efgtf  19823  funcrngcsetc  20776  funcrngcsetcALT  20777  funcringcsetc  20810  frlmphl  21968  pmatcollpw3lem  22977  mp2pm2mplem3  23002  txbas  23761  mpomulcn  25063  bcth  25525  opnmbl  25798  mbfimaopn  25852  mbfi1fseq  25917  om2noseqrdg  28534  noseqrdgsuc  28538  motplusg  28848  ttgval  29261  opsqrlem3  32531  elrgspnlem2  33594  splysubrg  33981  issply  33982  fedgmul  34052  mdetpmtr12  34246  madjusmdetlem4  34251  dya2iocival  34695  sxbrsigalem5  34710  sxbrsigalem6  34711  eulerpart  34804  sseqp1  34817  cvmliftlem15  35811  cvmlift2  35829  opnmbllem0  38348  mblfinlem1  38349  mblfinlem2  38350  sdc  38436  tendoplcbv  41590  dvhvaddcbv  41904  dvhvscacbv  41913  fsovcnvlem  44780  ntrneibex  44840  ioorrnopn  47060  hoidmvle  47355  ovnhoi  47358  hoimbl  47386  smflimlem6  47531  lmod1zr  49314  functhinclem4  50266