Theorem cbvmpov 7306

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5156, 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

Step	Hyp	Ref	Expression
1		nfcv 2904	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2904	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2904	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2904	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpov.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpov.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2798	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpo 7305	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1543   ∈ cmpo 7215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-oprab 7217  df-mpo 7218

This theorem is referenced by:  fvproj  7901  seqomlem0  8185  dffi3  9047  cantnfsuc  9285  fin23lem33  9959  om2uzrdg  13529  uzrdgsuci  13533  sadcp1  16014  smupp1  16039  imasvscafn  17042  mgmnsgrpex  18358  sgrpnmndex  18359  sylow1  18992  sylow2b  19012  sylow3lem5  19020  sylow3  19022  efgmval  19102  efgtf  19112  frlmphl  20743  pmatcollpw3lem  21680  mp2pm2mplem3  21705  txbas  22464  bcth  24226  opnmbl  24499  mbfimaopn  24553  mbfi1fseq  24619  motplusg  26633  ttgval  26966  opsqrlem3  30223  fedgmul  31426  mdetpmtr12  31489  madjusmdetlem4  31494  dya2iocival  31952  sxbrsigalem5  31967  sxbrsigalem6  31968  eulerpart  32061  sseqp1  32074  cvmliftlem15  32973  cvmlift2  32991  opnmbllem0  35550  mblfinlem1  35551  mblfinlem2  35552  sdc  35639  tendoplcbv  38526  dvhvaddcbv  38840  dvhvscacbv  38849  fsovcnvlem  41298  ntrneibex  41360  ioorrnopn  43521  hoidmvle  43813  ovnhoi  43816  hoimbl  43844  smflimlem6  43983  funcrngcsetc  45229  funcrngcsetcALT  45230  funcringcsetc  45266  lmod1zr  45507  functhinclem4  45998