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Theorem fmptcof 6884
Description: Version of fmptco 6883 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 3904 . . . . . . 7 𝑥𝑧 / 𝑥𝑅
32nfel1 2991 . . . . . 6 𝑥𝑧 / 𝑥𝑅𝐵
4 csbeq1a 3894 . . . . . . 7 (𝑥 = 𝑧𝑅 = 𝑧 / 𝑥𝑅)
54eleq1d 2894 . . . . . 6 (𝑥 = 𝑧 → (𝑅𝐵𝑧 / 𝑥𝑅𝐵))
63, 5rspc 3608 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 𝑅𝐵𝑧 / 𝑥𝑅𝐵))
71, 6mpan9 507 . . . 4 ((𝜑𝑧𝐴) → 𝑧 / 𝑥𝑅𝐵)
8 fmptcof.2 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝑅))
9 nfcv 2974 . . . . . 6 𝑧𝑅
109, 2, 4cbvmpt 5158 . . . . 5 (𝑥𝐴𝑅) = (𝑧𝐴𝑧 / 𝑥𝑅)
118, 10syl6eq 2869 . . . 4 (𝜑𝐹 = (𝑧𝐴𝑧 / 𝑥𝑅))
12 fmptcof.3 . . . . 5 (𝜑𝐺 = (𝑦𝐵𝑆))
13 nfcv 2974 . . . . . 6 𝑤𝑆
14 nfcsb1v 3904 . . . . . 6 𝑦𝑤 / 𝑦𝑆
15 csbeq1a 3894 . . . . . 6 (𝑦 = 𝑤𝑆 = 𝑤 / 𝑦𝑆)
1613, 14, 15cbvmpt 5158 . . . . 5 (𝑦𝐵𝑆) = (𝑤𝐵𝑤 / 𝑦𝑆)
1712, 16syl6eq 2869 . . . 4 (𝜑𝐺 = (𝑤𝐵𝑤 / 𝑦𝑆))
18 csbeq1 3883 . . . 4 (𝑤 = 𝑧 / 𝑥𝑅𝑤 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
197, 11, 17, 18fmptco 6883 . . 3 (𝜑 → (𝐺𝐹) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆))
20 nfcv 2974 . . . 4 𝑧𝑅 / 𝑦𝑆
21 nfcv 2974 . . . . 5 𝑥𝑆
222, 21nfcsbw 3906 . . . 4 𝑥𝑧 / 𝑥𝑅 / 𝑦𝑆
234csbeq1d 3884 . . . 4 (𝑥 = 𝑧𝑅 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
2420, 22, 23cbvmpt 5158 . . 3 (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆)
2519, 24syl6eqr 2871 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
26 eqid 2818 . . . 4 𝐴 = 𝐴
27 nfcvd 2975 . . . . . 6 (𝑅𝐵𝑦𝑇)
28 fmptcof.4 . . . . . 6 (𝑦 = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 3913 . . . . 5 (𝑅𝐵𝑅 / 𝑦𝑆 = 𝑇)
3029ralimi 3157 . . . 4 (∀𝑥𝐴 𝑅𝐵 → ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇)
31 mpteq12 5144 . . . 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 2853 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  csb 3880  cmpt 5137  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  fmptcos  6885  yonedalem3b  17517  gsumcom2  19024  evl1sca  20425  cnmptk1  22217  cnmpt1k  22218  cnmptkk  22219  cncfmpt1f  23448  copco  23549  pcoass  23555  sincn  24959  coscn  24960  lgseisenlem3  25880  fcomptf  30331  eulerpartgbij  31529  cncfcompt2  42058
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