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Theorem fmpt2co 7305
Description: Composition of two functions. Variation of fmptco 6436 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
fmpt2co.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)
fmpt2co.2 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
fmpt2co.3 (𝜑𝐺 = (𝑧𝐶𝑆))
fmpt2co.4 (𝑧 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmpt2co (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 fmpt2co
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmpt2co.1 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)
21ralrimivva 3000 . . . . 5 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑅𝐶)
3 eqid 2651 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑥𝐴, 𝑦𝐵𝑅)
43fmpt2 7282 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
52, 4sylib 208 . . . 4 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
6 nfcv 2793 . . . . . . 7 𝑢𝑅
7 nfcv 2793 . . . . . . 7 𝑣𝑅
8 nfcv 2793 . . . . . . . 8 𝑥𝑣
9 nfcsb1v 3582 . . . . . . . 8 𝑥𝑢 / 𝑥𝑅
108, 9nfcsb 3584 . . . . . . 7 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅
11 nfcsb1v 3582 . . . . . . 7 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅
12 csbeq1a 3575 . . . . . . . 8 (𝑥 = 𝑢𝑅 = 𝑢 / 𝑥𝑅)
13 csbeq1a 3575 . . . . . . . 8 (𝑦 = 𝑣𝑢 / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
1412, 13sylan9eq 2705 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
156, 7, 10, 11, 14cbvmpt2 6776 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
16 vex 3234 . . . . . . . . . 10 𝑢 ∈ V
17 vex 3234 . . . . . . . . . 10 𝑣 ∈ V
1816, 17op2ndd 7221 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) = 𝑣)
1918csbeq1d 3573 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅)
2016, 17op1std 7220 . . . . . . . . . 10 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) = 𝑢)
2120csbeq1d 3573 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) / 𝑥𝑅 = 𝑢 / 𝑥𝑅)
2221csbeq2dv 4025 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2319, 22eqtrd 2685 . . . . . . 7 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2423mpt2mpt 6794 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
2515, 24eqtr4i 2676 . . . . 5 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅)
2625fmpt 6421 . . . 4 (∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
275, 26sylibr 224 . . 3 (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶)
28 fmpt2co.2 . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
2928, 25syl6eq 2701 . . 3 (𝜑𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅))
30 fmpt2co.3 . . 3 (𝜑𝐺 = (𝑧𝐶𝑆))
3127, 29, 30fmptcos 6438 . 2 (𝜑 → (𝐺𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆))
3223csbeq1d 3573 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
3332mpt2mpt 6794 . . . 4 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
34 nfcv 2793 . . . . 5 𝑢𝑅 / 𝑧𝑆
35 nfcv 2793 . . . . 5 𝑣𝑅 / 𝑧𝑆
36 nfcv 2793 . . . . . 6 𝑥𝑆
3710, 36nfcsb 3584 . . . . 5 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
38 nfcv 2793 . . . . . 6 𝑦𝑆
3911, 38nfcsb 3584 . . . . 5 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
4014csbeq1d 3573 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4134, 35, 37, 39, 40cbvmpt2 6776 . . . 4 (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4233, 41eqtr4i 2676 . . 3 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆)
4313impb 1279 . . . . 5 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅𝐶)
44 nfcvd 2794 . . . . . 6 (𝑅𝐶𝑧𝑇)
45 fmpt2co.4 . . . . . 6 (𝑧 = 𝑅𝑆 = 𝑇)
4644, 45csbiegf 3590 . . . . 5 (𝑅𝐶𝑅 / 𝑧𝑆 = 𝑇)
4743, 46syl 17 . . . 4 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅 / 𝑧𝑆 = 𝑇)
4847mpt2eq3dva 6761 . . 3 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
4942, 48syl5eq 2697 . 2 (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
5031, 49eqtrd 2685 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  csb 3566  cop 4216  cmpt 4762   × cxp 5141  ccom 5147  wf 5922  cfv 5926  cmpt2 6692  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  oprabco  7306  evlslem2  19560  txswaphmeolem  21655  xpstopnlem1  21660  stdbdxmet  22367  rrxds  23227  cnre2csqima  30085  cvmlift2lem7  31417
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