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Theorem fmptcof 6561
Description: Version of fmptco 6560 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 3690 . . . . . . 7 𝑥𝑧 / 𝑥𝑅
32nfel1 2917 . . . . . 6 𝑥𝑧 / 𝑥𝑅𝐵
4 csbeq1a 3683 . . . . . . 7 (𝑥 = 𝑧𝑅 = 𝑧 / 𝑥𝑅)
54eleq1d 2824 . . . . . 6 (𝑥 = 𝑧 → (𝑅𝐵𝑧 / 𝑥𝑅𝐵))
63, 5rspc 3443 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 𝑅𝐵𝑧 / 𝑥𝑅𝐵))
71, 6mpan9 487 . . . 4 ((𝜑𝑧𝐴) → 𝑧 / 𝑥𝑅𝐵)
8 fmptcof.2 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝑅))
9 nfcv 2902 . . . . . 6 𝑧𝑅
109, 2, 4cbvmpt 4901 . . . . 5 (𝑥𝐴𝑅) = (𝑧𝐴𝑧 / 𝑥𝑅)
118, 10syl6eq 2810 . . . 4 (𝜑𝐹 = (𝑧𝐴𝑧 / 𝑥𝑅))
12 fmptcof.3 . . . . 5 (𝜑𝐺 = (𝑦𝐵𝑆))
13 nfcv 2902 . . . . . 6 𝑤𝑆
14 nfcsb1v 3690 . . . . . 6 𝑦𝑤 / 𝑦𝑆
15 csbeq1a 3683 . . . . . 6 (𝑦 = 𝑤𝑆 = 𝑤 / 𝑦𝑆)
1613, 14, 15cbvmpt 4901 . . . . 5 (𝑦𝐵𝑆) = (𝑤𝐵𝑤 / 𝑦𝑆)
1712, 16syl6eq 2810 . . . 4 (𝜑𝐺 = (𝑤𝐵𝑤 / 𝑦𝑆))
18 csbeq1 3677 . . . 4 (𝑤 = 𝑧 / 𝑥𝑅𝑤 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
197, 11, 17, 18fmptco 6560 . . 3 (𝜑 → (𝐺𝐹) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆))
20 nfcv 2902 . . . 4 𝑧𝑅 / 𝑦𝑆
21 nfcv 2902 . . . . 5 𝑥𝑆
222, 21nfcsb 3692 . . . 4 𝑥𝑧 / 𝑥𝑅 / 𝑦𝑆
234csbeq1d 3681 . . . 4 (𝑥 = 𝑧𝑅 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
2420, 22, 23cbvmpt 4901 . . 3 (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆)
2519, 24syl6eqr 2812 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
26 eqid 2760 . . . 4 𝐴 = 𝐴
27 nfcvd 2903 . . . . . 6 (𝑅𝐵𝑦𝑇)
28 fmptcof.4 . . . . . 6 (𝑦 = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 3698 . . . . 5 (𝑅𝐵𝑅 / 𝑦𝑆 = 𝑇)
3029ralimi 3090 . . . 4 (∀𝑥𝐴 𝑅𝐵 → ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇)
31 mpteq12 4888 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇) → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3226, 30, 31sylancr 698 . . 3 (∀𝑥𝐴 𝑅𝐵 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
331, 32syl 17 . 2 (𝜑 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3425, 33eqtrd 2794 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  csb 3674  cmpt 4881  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  fmptcos  6562  yonedalem3b  17140  gsumcom2  18594  evl1sca  19920  cnmptk1  21706  cnmpt1k  21707  cnmptkk  21708  cncfmpt1f  22937  copco  23038  pcoass  23044  sincn  24417  coscn  24418  lgseisenlem3  25322  fcomptf  29788  eulerpartgbij  30764  cncfcompt2  40633
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