MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpoco Structured version   Visualization version   GIF version

Theorem fmpoco 8076
Description: Composition of two functions. Variation of fmptco 7120 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
fmpoco.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)
fmpoco.2 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
fmpoco.3 (𝜑𝐺 = (𝑧𝐶𝑆))
fmpoco.4 (𝑧 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmpoco (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑧,𝑅   𝑧,𝑇
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝑅(𝑥,𝑦)   𝑆(𝑧)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem fmpoco
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmpoco.1 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)
21ralrimivva 3192 . . . . 5 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑅𝐶)
3 eqid 2724 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑥𝐴, 𝑦𝐵𝑅)
43fmpo 8048 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
52, 4sylib 217 . . . 4 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
6 nfcv 2895 . . . . . . 7 𝑢𝑅
7 nfcv 2895 . . . . . . 7 𝑣𝑅
8 nfcv 2895 . . . . . . . 8 𝑥𝑣
9 nfcsb1v 3911 . . . . . . . 8 𝑥𝑢 / 𝑥𝑅
108, 9nfcsbw 3913 . . . . . . 7 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅
11 nfcsb1v 3911 . . . . . . 7 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅
12 csbeq1a 3900 . . . . . . . 8 (𝑥 = 𝑢𝑅 = 𝑢 / 𝑥𝑅)
13 csbeq1a 3900 . . . . . . . 8 (𝑦 = 𝑣𝑢 / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
1412, 13sylan9eq 2784 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
156, 7, 10, 11, 14cbvmpo 7496 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
16 vex 3470 . . . . . . . . . 10 𝑢 ∈ V
17 vex 3470 . . . . . . . . . 10 𝑣 ∈ V
1816, 17op2ndd 7980 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) = 𝑣)
1918csbeq1d 3890 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅)
2016, 17op1std 7979 . . . . . . . . . 10 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) = 𝑢)
2120csbeq1d 3890 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) / 𝑥𝑅 = 𝑢 / 𝑥𝑅)
2221csbeq2dv 3893 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2319, 22eqtrd 2764 . . . . . . 7 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2423mpompt 7515 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
2515, 24eqtr4i 2755 . . . . 5 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅)
2625fmpt 7102 . . . 4 (∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
275, 26sylibr 233 . . 3 (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶)
28 fmpoco.2 . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
2928, 25eqtrdi 2780 . . 3 (𝜑𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅))
30 fmpoco.3 . . 3 (𝜑𝐺 = (𝑧𝐶𝑆))
3127, 29, 30fmptcos 7122 . 2 (𝜑 → (𝐺𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆))
3223csbeq1d 3890 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
3332mpompt 7515 . . . 4 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
34 nfcv 2895 . . . . 5 𝑢𝑅 / 𝑧𝑆
35 nfcv 2895 . . . . 5 𝑣𝑅 / 𝑧𝑆
36 nfcv 2895 . . . . . 6 𝑥𝑆
3710, 36nfcsbw 3913 . . . . 5 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
38 nfcv 2895 . . . . . 6 𝑦𝑆
3911, 38nfcsbw 3913 . . . . 5 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
4014csbeq1d 3890 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4134, 35, 37, 39, 40cbvmpo 7496 . . . 4 (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4233, 41eqtr4i 2755 . . 3 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆)
4313impb 1112 . . . . 5 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅𝐶)
44 nfcvd 2896 . . . . . 6 (𝑅𝐶𝑧𝑇)
45 fmpoco.4 . . . . . 6 (𝑧 = 𝑅𝑆 = 𝑇)
4644, 45csbiegf 3920 . . . . 5 (𝑅𝐶𝑅 / 𝑧𝑆 = 𝑇)
4743, 46syl 17 . . . 4 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅 / 𝑧𝑆 = 𝑇)
4847mpoeq3dva 7479 . . 3 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
4942, 48eqtrid 2776 . 2 (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
5031, 49eqtrd 2764 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  csb 3886  cop 4627  cmpt 5222   × cxp 5665  ccom 5671  wf 6530  cfv 6534  cmpo 7404  1st c1st 7967  2nd c2nd 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970
This theorem is referenced by:  oprabco  8077  evlslem2  21954  txswaphmeolem  23632  xpstopnlem1  23637  stdbdxmet  24348  rrxds  25245  cnre2csqima  33383  cvmlift2lem7  34791
  Copyright terms: Public domain W3C validator