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Theorem fmptcof 5449
Description: Version of fmptco 5448 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 2961 . . . . . . 7 𝑥𝑧 / 𝑥𝑅
32nfel1 2239 . . . . . 6 𝑥𝑧 / 𝑥𝑅𝐵
4 csbeq1a 2939 . . . . . . 7 (𝑥 = 𝑧𝑅 = 𝑧 / 𝑥𝑅)
54eleq1d 2156 . . . . . 6 (𝑥 = 𝑧 → (𝑅𝐵𝑧 / 𝑥𝑅𝐵))
63, 5rspc 2716 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 𝑅𝐵𝑧 / 𝑥𝑅𝐵))
71, 6mpan9 275 . . . 4 ((𝜑𝑧𝐴) → 𝑧 / 𝑥𝑅𝐵)
8 fmptcof.2 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝑅))
9 nfcv 2228 . . . . . 6 𝑧𝑅
109, 2, 4cbvmpt 3925 . . . . 5 (𝑥𝐴𝑅) = (𝑧𝐴𝑧 / 𝑥𝑅)
118, 10syl6eq 2136 . . . 4 (𝜑𝐹 = (𝑧𝐴𝑧 / 𝑥𝑅))
12 fmptcof.3 . . . . 5 (𝜑𝐺 = (𝑦𝐵𝑆))
13 nfcv 2228 . . . . . 6 𝑤𝑆
14 nfcsb1v 2961 . . . . . 6 𝑦𝑤 / 𝑦𝑆
15 csbeq1a 2939 . . . . . 6 (𝑦 = 𝑤𝑆 = 𝑤 / 𝑦𝑆)
1613, 14, 15cbvmpt 3925 . . . . 5 (𝑦𝐵𝑆) = (𝑤𝐵𝑤 / 𝑦𝑆)
1712, 16syl6eq 2136 . . . 4 (𝜑𝐺 = (𝑤𝐵𝑤 / 𝑦𝑆))
18 csbeq1 2934 . . . 4 (𝑤 = 𝑧 / 𝑥𝑅𝑤 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
197, 11, 17, 18fmptco 5448 . . 3 (𝜑 → (𝐺𝐹) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆))
20 nfcv 2228 . . . 4 𝑧𝑅 / 𝑦𝑆
21 nfcv 2228 . . . . 5 𝑥𝑆
222, 21nfcsb 2963 . . . 4 𝑥𝑧 / 𝑥𝑅 / 𝑦𝑆
234csbeq1d 2937 . . . 4 (𝑥 = 𝑧𝑅 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
2420, 22, 23cbvmpt 3925 . . 3 (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆)
2519, 24syl6eqr 2138 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
26 eqid 2088 . . . 4 𝐴 = 𝐴
27 nfcvd 2229 . . . . . 6 (𝑅𝐵𝑦𝑇)
28 fmptcof.4 . . . . . 6 (𝑦 = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 2969 . . . . 5 (𝑅𝐵𝑅 / 𝑦𝑆 = 𝑇)
3029ralimi 2438 . . . 4 (∀𝑥𝐴 𝑅𝐵 → ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇)
31 mpteq12 3913 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇) → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3226, 30, 31sylancr 405 . . 3 (∀𝑥𝐴 𝑅𝐵 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
331, 32syl 14 . 2 (𝜑 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3425, 33eqtrd 2120 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  wral 2359  csb 2931  cmpt 3891  ccom 4432
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
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 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010
This theorem is referenced by:  fmptcos  5450
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