ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmptcof GIF version

Theorem fmptcof 5683
Description: Version of fmptco 5682 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 3090 . . . . . . 7 𝑥𝑧 / 𝑥𝑅
32nfel1 2330 . . . . . 6 𝑥𝑧 / 𝑥𝑅𝐵
4 csbeq1a 3066 . . . . . . 7 (𝑥 = 𝑧𝑅 = 𝑧 / 𝑥𝑅)
54eleq1d 2246 . . . . . 6 (𝑥 = 𝑧 → (𝑅𝐵𝑧 / 𝑥𝑅𝐵))
63, 5rspc 2835 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 𝑅𝐵𝑧 / 𝑥𝑅𝐵))
71, 6mpan9 281 . . . 4 ((𝜑𝑧𝐴) → 𝑧 / 𝑥𝑅𝐵)
8 fmptcof.2 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝑅))
9 nfcv 2319 . . . . . 6 𝑧𝑅
109, 2, 4cbvmpt 4098 . . . . 5 (𝑥𝐴𝑅) = (𝑧𝐴𝑧 / 𝑥𝑅)
118, 10eqtrdi 2226 . . . 4 (𝜑𝐹 = (𝑧𝐴𝑧 / 𝑥𝑅))
12 fmptcof.3 . . . . 5 (𝜑𝐺 = (𝑦𝐵𝑆))
13 nfcv 2319 . . . . . 6 𝑤𝑆
14 nfcsb1v 3090 . . . . . 6 𝑦𝑤 / 𝑦𝑆
15 csbeq1a 3066 . . . . . 6 (𝑦 = 𝑤𝑆 = 𝑤 / 𝑦𝑆)
1613, 14, 15cbvmpt 4098 . . . . 5 (𝑦𝐵𝑆) = (𝑤𝐵𝑤 / 𝑦𝑆)
1712, 16eqtrdi 2226 . . . 4 (𝜑𝐺 = (𝑤𝐵𝑤 / 𝑦𝑆))
18 csbeq1 3060 . . . 4 (𝑤 = 𝑧 / 𝑥𝑅𝑤 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
197, 11, 17, 18fmptco 5682 . . 3 (𝜑 → (𝐺𝐹) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆))
20 nfcv 2319 . . . 4 𝑧𝑅 / 𝑦𝑆
21 nfcv 2319 . . . . 5 𝑥𝑆
222, 21nfcsb 3094 . . . 4 𝑥𝑧 / 𝑥𝑅 / 𝑦𝑆
234csbeq1d 3064 . . . 4 (𝑥 = 𝑧𝑅 / 𝑦𝑆 = 𝑧 / 𝑥𝑅 / 𝑦𝑆)
2420, 22, 23cbvmpt 4098 . . 3 (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑧𝐴𝑧 / 𝑥𝑅 / 𝑦𝑆)
2519, 24eqtr4di 2228 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
26 eqid 2177 . . . 4 𝐴 = 𝐴
27 nfcvd 2320 . . . . . 6 (𝑅𝐵𝑦𝑇)
28 fmptcof.4 . . . . . 6 (𝑦 = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 3100 . . . . 5 (𝑅𝐵𝑅 / 𝑦𝑆 = 𝑇)
3029ralimi 2540 . . . 4 (∀𝑥𝐴 𝑅𝐵 → ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇)
31 mpteq12 4086 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝑅 / 𝑦𝑆 = 𝑇) → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3226, 30, 31sylancr 414 . . 3 (∀𝑥𝐴 𝑅𝐵 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
331, 32syl 14 . 2 (𝜑 → (𝑥𝐴𝑅 / 𝑦𝑆) = (𝑥𝐴𝑇))
3425, 33eqtrd 2210 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  csb 3057  cmpt 4064  ccom 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224
This theorem is referenced by:  fmptcos  5684  cncfmpt1f  13977  sincn  14083  coscn  14084
  Copyright terms: Public domain W3C validator