Theorem cbvmpov 7487

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5212, 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 2812	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		eleq1w 2812	. . . . 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 2785	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
7	6	eqeq2d 2741	. . . 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 7482	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
10		df-mpo 7395	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
11		df-mpo 7395	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 = 𝐷)}
12	9, 10, 11	3eqtr4i 2763	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {coprab 7391   ∈ cmpo 7392

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  fvproj  8116  seqomlem0  8420  dffi3  9389  cantnfsuc  9630  fin23lem33  10305  om2uzrdg  13928  uzrdgsuci  13932  sadcp1  16432  smupp1  16457  imasvscafn  17507  mgmnsgrpex  18865  sgrpnmndex  18866  sylow1  19540  sylow2b  19560  sylow3lem5  19568  sylow3  19570  efgmval  19649  efgtf  19659  funcrngcsetc  20556  funcrngcsetcALT  20557  funcringcsetc  20590  frlmphl  21697  pmatcollpw3lem  22677  mp2pm2mplem3  22702  txbas  23461  mpomulcn  24765  bcth  25236  opnmbl  25510  mbfimaopn  25564  mbfi1fseq  25629  om2noseqrdg  28205  noseqrdgsuc  28209  motplusg  28476  ttgval  28809  opsqrlem3  32078  elrgspnlem2  33201  fedgmul  33634  mdetpmtr12  33822  madjusmdetlem4  33827  dya2iocival  34271  sxbrsigalem5  34286  sxbrsigalem6  34287  eulerpart  34380  sseqp1  34393  cvmliftlem15  35292  cvmlift2  35310  opnmbllem0  37657  mblfinlem1  37658  mblfinlem2  37659  sdc  37745  tendoplcbv  40776  dvhvaddcbv  41090  dvhvscacbv  41099  fsovcnvlem  44009  ntrneibex  44069  ioorrnopn  46310  hoidmvle  46605  ovnhoi  46608  hoimbl  46636  smflimlem6  46781  lmod1zr  48486  functhinclem4  49440