Theorem cbvmpov 7448

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5197, 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 2811	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		eleq1w 2811	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
3	1, 2	bi2anan9 638	. . . 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 2784	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
7	6	eqeq2d 2740	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = 𝐷))
8	3, 7	anbi12d 632	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)))
9	8	cbvoprab12v 7443	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
10		df-mpo 7358	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
11		df-mpo 7358	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
12	9, 10, 11	3eqtr4i 2762	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {coprab 7354   ∈ cmpo 7355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-oprab 7357  df-mpo 7358

This theorem is referenced by:  fvproj  8074  seqomlem0  8378  dffi3  9340  cantnfsuc  9585  fin23lem33  10258  om2uzrdg  13881  uzrdgsuci  13885  sadcp1  16384  smupp1  16409  imasvscafn  17459  mgmnsgrpex  18823  sgrpnmndex  18824  sylow1  19500  sylow2b  19520  sylow3lem5  19528  sylow3  19530  efgmval  19609  efgtf  19619  funcrngcsetc  20543  funcrngcsetcALT  20544  funcringcsetc  20577  frlmphl  21706  pmatcollpw3lem  22686  mp2pm2mplem3  22711  txbas  23470  mpomulcn  24774  bcth  25245  opnmbl  25519  mbfimaopn  25573  mbfi1fseq  25638  om2noseqrdg  28221  noseqrdgsuc  28225  motplusg  28505  ttgval  28838  opsqrlem3  32104  elrgspnlem2  33193  fedgmul  33603  mdetpmtr12  33791  madjusmdetlem4  33796  dya2iocival  34240  sxbrsigalem5  34255  sxbrsigalem6  34256  eulerpart  34349  sseqp1  34362  cvmliftlem15  35270  cvmlift2  35288  opnmbllem0  37635  mblfinlem1  37636  mblfinlem2  37637  sdc  37723  tendoplcbv  40754  dvhvaddcbv  41068  dvhvscacbv  41077  fsovcnvlem  43986  ntrneibex  44046  ioorrnopn  46287  hoidmvle  46582  ovnhoi  46585  hoimbl  46613  smflimlem6  46758  lmod1zr  48479  functhinclem4  49433