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Theorem fmptcof 7150
Description: Version of fmptco 7149 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmptcof.1 (𝜑 → ∀𝑥𝐴 𝑅𝐵)
fmptcof.2 (𝜑𝐹 = (𝑥𝐴𝑅))
fmptcof.3 (𝜑𝐺 = (𝑦𝐵𝑆))
fmptcof.4 (𝑦 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmptcof (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑦,𝑅   𝑥,𝑆   𝑥,𝐴   𝑦,𝑇
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑇(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fmptcof
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmptcof.1 . . . . 5 (𝜑 → ∀𝑥𝐴 𝑅𝐵)
2 nfcsb1v 3923 . . . . . . 7 𝑥𝑧 / 𝑥𝑅
32nfel1 2922 . . . . . 6 𝑥𝑧 / 𝑥𝑅𝐵
4 csbeq1a 3913 . . . . . . 7 (𝑥 = 𝑧𝑅 = 𝑧 / 𝑥𝑅)
54eleq1d 2826 . . . . . 6 (𝑥 = 𝑧 → (𝑅𝐵𝑧 / 𝑥𝑅𝐵))
63, 5rspc 3610 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 𝑅𝐵𝑧 / 𝑥𝑅𝐵))
71, 6mpan9 506 . . . 4 ((𝜑𝑧𝐴) → 𝑧 / 𝑥𝑅𝐵)
8 fmptcof.2 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝑅))
9 nfcv 2905 . . . . . 6 𝑧𝑅
109, 2, 4cbvmpt 5253 . . . . 5 (𝑥𝐴𝑅) = (𝑧𝐴𝑧 / 𝑥𝑅)
118, 10eqtrdi 2793 . . . 4 (𝜑𝐹 = (𝑧𝐴𝑧 / 𝑥𝑅))
12 fmptcof.3 . . . . 5 (𝜑𝐺 = (𝑦𝐵𝑆))
13 nfcv 2905 . . . . . 6 𝑤𝑆
14 nfcsb1v 3923 . . . . . 6 𝑦𝑤 / 𝑦𝑆
15 csbeq1a 3913 . . . . . 6 (𝑦 = 𝑤𝑆 = 𝑤 / 𝑦𝑆)
1613, 14, 15cbvmpt 5253 . . . . 5 (𝑦𝐵𝑆) = (𝑤𝐵𝑤 / 𝑦𝑆)
1712, 16eqtrdi 2793 . . . 4 (𝜑𝐺 = (𝑤𝐵𝑤 / 𝑦𝑆))
18 csbeq1 3902 . . . 4 (𝑤 = 𝑧 / 𝑥𝑅𝑤 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
197, 11, 17, 18fmptco 7149 . . 3 (𝜑 → (𝐺𝐹) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆))
20 nfcv 2905 . . . 4 𝑧𝑅 / 𝑦𝑆
21 nfcv 2905 . . . . 5 𝑥𝑆
222, 21nfcsbw 3925 . . . 4 𝑥𝑧 / 𝑥𝑅 / 𝑦𝑆
234csbeq1d 3903 . . . 4 (𝑥 = 𝑧𝑅 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
2420, 22, 23cbvmpt 5253 . . 3 (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆)
2519, 24eqtr4di 2795 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
26 eqid 2737 . . . 4 𝐴 = 𝐴
27 nfcvd 2906 . . . . . 6 (𝑅𝐵𝑦𝑇)
28 fmptcof.4 . . . . . 6 (𝑦 = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 3932 . . . . 5 (𝑅𝐵𝑅 / 𝑦𝑆 = 𝑇)
3029ralimi 3083 . . . 4 (∀𝑥𝐴 𝑅𝐵 → ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇)
31 mpteq12 5234 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇) → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3226, 30, 31sylancr 587 . . 3 (∀𝑥𝐴 𝑅𝐵 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
331, 32syl 17 . 2 (𝜑 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3425, 33eqtrd 2777 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  csb 3899  cmpt 5225  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  fmptcos  7151  yonedalem3b  18324  gsumcom2  19993  evl1sca  22338  cnmptk1  23689  cnmpt1k  23690  cnmptkk  23691  cncfcompt2  24934  cncfmpt1f  24940  copco  25051  pcoass  25057  sincn  26488  coscn  26489  lgseisenlem3  27421  fcomptf  32668  eulerpartgbij  34374
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