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Theorem fmpt2co 5981
Description: Composition of two functions. Variation of fmptco 5464 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 2455 . . . . 5 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑅𝐶)
3 eqid 2088 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑥𝐴, 𝑦𝐵𝑅)
43fmpt2 5971 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
52, 4sylib 120 . . . 4 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
6 nfcv 2228 . . . . . . 7 𝑢𝑅
7 nfcv 2228 . . . . . . 7 𝑣𝑅
8 nfcv 2228 . . . . . . . 8 𝑥𝑣
9 nfcsb1v 2963 . . . . . . . 8 𝑥𝑢 / 𝑥𝑅
108, 9nfcsb 2965 . . . . . . 7 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅
11 nfcsb1v 2963 . . . . . . 7 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅
12 csbeq1a 2941 . . . . . . . 8 (𝑥 = 𝑢𝑅 = 𝑢 / 𝑥𝑅)
13 csbeq1a 2941 . . . . . . . 8 (𝑦 = 𝑣𝑢 / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
1412, 13sylan9eq 2140 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
156, 7, 10, 11, 14cbvmpt2 5727 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
16 vex 2622 . . . . . . . . . 10 𝑢 ∈ V
17 vex 2622 . . . . . . . . . 10 𝑣 ∈ V
1816, 17op2ndd 5920 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) = 𝑣)
1918csbeq1d 2939 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅)
2016, 17op1std 5919 . . . . . . . . . 10 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) = 𝑢)
2120csbeq1d 2939 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) / 𝑥𝑅 = 𝑢 / 𝑥𝑅)
2221csbeq2dv 2956 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2319, 22eqtrd 2120 . . . . . . 7 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2423mpt2mpt 5740 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
2515, 24eqtr4i 2111 . . . . 5 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅)
2625fmpt 5449 . . . 4 (∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
275, 26sylibr 132 . . 3 (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶)
28 fmpt2co.2 . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
2928, 25syl6eq 2136 . . 3 (𝜑𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅))
30 fmpt2co.3 . . 3 (𝜑𝐺 = (𝑧𝐶𝑆))
3127, 29, 30fmptcos 5466 . 2 (𝜑 → (𝐺𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆))
3223csbeq1d 2939 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
3332mpt2mpt 5740 . . . 4 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
34 nfcv 2228 . . . . 5 𝑢𝑅 / 𝑧𝑆
35 nfcv 2228 . . . . 5 𝑣𝑅 / 𝑧𝑆
36 nfcv 2228 . . . . . 6 𝑥𝑆
3710, 36nfcsb 2965 . . . . 5 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
38 nfcv 2228 . . . . . 6 𝑦𝑆
3911, 38nfcsb 2965 . . . . 5 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
4014csbeq1d 2939 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4134, 35, 37, 39, 40cbvmpt2 5727 . . . 4 (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4233, 41eqtr4i 2111 . . 3 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆)
4313impb 1139 . . . . 5 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅𝐶)
44 nfcvd 2229 . . . . . 6 (𝑅𝐶𝑧𝑇)
45 fmpt2co.4 . . . . . 6 (𝑧 = 𝑅𝑆 = 𝑇)
4644, 45csbiegf 2971 . . . . 5 (𝑅𝐶𝑅 / 𝑧𝑆 = 𝑇)
4743, 46syl 14 . . . 4 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅 / 𝑧𝑆 = 𝑇)
4847mpt2eq3dva 5713 . . 3 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
4942, 48syl5eq 2132 . 2 (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
5031, 49eqtrd 2120 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  wral 2359  csb 2933  cop 3449  cmpt 3899   × cxp 4436  ccom 4442  wf 5011  cfv 5015  cmpt2 5654  1st c1st 5909  2nd c2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912
This theorem is referenced by:  oprabco  5982
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