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Theorem fmpoco 7816
Description: Composition of two functions. Variation of fmptco 6901 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 3103 . . . . 5 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑅𝐶)
3 eqid 2738 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑥𝐴, 𝑦𝐵𝑅)
43fmpo 7791 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
52, 4sylib 221 . . . 4 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
6 nfcv 2899 . . . . . . 7 𝑢𝑅
7 nfcv 2899 . . . . . . 7 𝑣𝑅
8 nfcv 2899 . . . . . . . 8 𝑥𝑣
9 nfcsb1v 3814 . . . . . . . 8 𝑥𝑢 / 𝑥𝑅
108, 9nfcsbw 3816 . . . . . . 7 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅
11 nfcsb1v 3814 . . . . . . 7 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅
12 csbeq1a 3804 . . . . . . . 8 (𝑥 = 𝑢𝑅 = 𝑢 / 𝑥𝑅)
13 csbeq1a 3804 . . . . . . . 8 (𝑦 = 𝑣𝑢 / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
1412, 13sylan9eq 2793 . . . . . . 7 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
156, 7, 10, 11, 14cbvmpo 7262 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
16 vex 3402 . . . . . . . . . 10 𝑢 ∈ V
17 vex 3402 . . . . . . . . . 10 𝑣 ∈ V
1816, 17op2ndd 7725 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) = 𝑣)
1918csbeq1d 3794 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅)
2016, 17op1std 7724 . . . . . . . . . 10 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) = 𝑢)
2120csbeq1d 3794 . . . . . . . . 9 (𝑤 = ⟨𝑢, 𝑣⟩ → (1st𝑤) / 𝑥𝑅 = 𝑢 / 𝑥𝑅)
2221csbeq2dv 3797 . . . . . . . 8 (𝑤 = ⟨𝑢, 𝑣⟩ → 𝑣 / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2319, 22eqtrd 2773 . . . . . . 7 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 = 𝑣 / 𝑦𝑢 / 𝑥𝑅)
2423mpompt 7280 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅)
2515, 24eqtr4i 2764 . . . . 5 (𝑥𝐴, 𝑦𝐵𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅)
2625fmpt 6884 . . . 4 (∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶 ↔ (𝑥𝐴, 𝑦𝐵𝑅):(𝐴 × 𝐵)⟶𝐶)
275, 26sylibr 237 . . 3 (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)(2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅𝐶)
28 fmpoco.2 . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))
2928, 25eqtrdi 2789 . . 3 (𝜑𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅))
30 fmpoco.3 . . 3 (𝜑𝐺 = (𝑧𝐶𝑆))
3127, 29, 30fmptcos 6903 . 2 (𝜑 → (𝐺𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆))
3223csbeq1d 3794 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
3332mpompt 7280 . . . 4 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
34 nfcv 2899 . . . . 5 𝑢𝑅 / 𝑧𝑆
35 nfcv 2899 . . . . 5 𝑣𝑅 / 𝑧𝑆
36 nfcv 2899 . . . . . 6 𝑥𝑆
3710, 36nfcsbw 3816 . . . . 5 𝑥𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
38 nfcv 2899 . . . . . 6 𝑦𝑆
3911, 38nfcsbw 3816 . . . . 5 𝑦𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆
4014csbeq1d 3794 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑅 / 𝑧𝑆 = 𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4134, 35, 37, 39, 40cbvmpo 7262 . . . 4 (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑢𝐴, 𝑣𝐵𝑣 / 𝑦𝑢 / 𝑥𝑅 / 𝑧𝑆)
4233, 41eqtr4i 2764 . . 3 (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆)
4313impb 1116 . . . . 5 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅𝐶)
44 nfcvd 2900 . . . . . 6 (𝑅𝐶𝑧𝑇)
45 fmpoco.4 . . . . . 6 (𝑧 = 𝑅𝑆 = 𝑇)
4644, 45csbiegf 3823 . . . . 5 (𝑅𝐶𝑅 / 𝑧𝑆 = 𝑇)
4743, 46syl 17 . . . 4 ((𝜑𝑥𝐴𝑦𝐵) → 𝑅 / 𝑧𝑆 = 𝑇)
4847mpoeq3dva 7245 . . 3 (𝜑 → (𝑥𝐴, 𝑦𝐵𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
4942, 48syl5eq 2785 . 2 (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ (2nd𝑤) / 𝑦(1st𝑤) / 𝑥𝑅 / 𝑧𝑆) = (𝑥𝐴, 𝑦𝐵𝑇))
5031, 49eqtrd 2773 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  csb 3790  cop 4522  cmpt 5110   × cxp 5523  ccom 5529  wf 6335  cfv 6339  cmpo 7172  1st c1st 7712  2nd c2nd 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715
This theorem is referenced by:  oprabco  7817  evlslem2  20893  txswaphmeolem  22555  xpstopnlem1  22560  stdbdxmet  23268  rrxds  24145  cnre2csqima  31433  cvmlift2lem7  32842
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