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

Theorem fmptcof 5274
Description: Version of fmptco 5273 where φ needn't be distinct from x. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmptcof.1 (φx A 𝑅 B)
fmptcof.2 (φ𝐹 = (x A𝑅))
fmptcof.3 (φ𝐺 = (y B𝑆))
fmptcof.4 (y = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmptcof (φ → (𝐺𝐹) = (x A𝑇))
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 (φx A 𝑅 B)
2 nfcsb1v 2876 . . . . . . 7 xz / x𝑅
32nfel1 2185 . . . . . 6 xz / x𝑅 B
4 csbeq1a 2854 . . . . . . 7 (x = z𝑅 = z / x𝑅)
54eleq1d 2103 . . . . . 6 (x = z → (𝑅 Bz / x𝑅 B))
63, 5rspc 2644 . . . . 5 (z A → (x A 𝑅 Bz / x𝑅 B))
71, 6mpan9 265 . . . 4 ((φ z A) → z / x𝑅 B)
8 fmptcof.2 . . . . 5 (φ𝐹 = (x A𝑅))
9 nfcv 2175 . . . . . 6 z𝑅
109, 2, 4cbvmpt 3842 . . . . 5 (x A𝑅) = (z Az / x𝑅)
118, 10syl6eq 2085 . . . 4 (φ𝐹 = (z Az / x𝑅))
12 fmptcof.3 . . . . 5 (φ𝐺 = (y B𝑆))
13 nfcv 2175 . . . . . 6 w𝑆
14 nfcsb1v 2876 . . . . . 6 yw / y𝑆
15 csbeq1a 2854 . . . . . 6 (y = w𝑆 = w / y𝑆)
1613, 14, 15cbvmpt 3842 . . . . 5 (y B𝑆) = (w Bw / y𝑆)
1712, 16syl6eq 2085 . . . 4 (φ𝐺 = (w Bw / y𝑆))
18 csbeq1 2849 . . . 4 (w = z / x𝑅w / y𝑆 = z / x𝑅 / y𝑆)
197, 11, 17, 18fmptco 5273 . . 3 (φ → (𝐺𝐹) = (z Az / x𝑅 / y𝑆))
20 nfcv 2175 . . . 4 z𝑅 / y𝑆
21 nfcv 2175 . . . . 5 x𝑆
222, 21nfcsb 2878 . . . 4 xz / x𝑅 / y𝑆
234csbeq1d 2852 . . . 4 (x = z𝑅 / y𝑆 = z / x𝑅 / y𝑆)
2420, 22, 23cbvmpt 3842 . . 3 (x A𝑅 / y𝑆) = (z Az / x𝑅 / y𝑆)
2519, 24syl6eqr 2087 . 2 (φ → (𝐺𝐹) = (x A𝑅 / y𝑆))
26 eqid 2037 . . . 4 A = A
27 nfcvd 2176 . . . . . 6 (𝑅 By𝑇)
28 fmptcof.4 . . . . . 6 (y = 𝑅𝑆 = 𝑇)
2927, 28csbiegf 2884 . . . . 5 (𝑅 B𝑅 / y𝑆 = 𝑇)
3029ralimi 2378 . . . 4 (x A 𝑅 Bx A 𝑅 / y𝑆 = 𝑇)
31 mpteq12 3831 . . . 4 ((A = A x A 𝑅 / y𝑆 = 𝑇) → (x A𝑅 / y𝑆) = (x A𝑇))
3226, 30, 31sylancr 393 . . 3 (x A 𝑅 B → (x A𝑅 / y𝑆) = (x A𝑇))
331, 32syl 14 . 2 (φ → (x A𝑅 / y𝑆) = (x A𝑇))
3425, 33eqtrd 2069 1 (φ → (𝐺𝐹) = (x A𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wral 2300  csb 2846  cmpt 3809  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by:  fmptcos  5275
  Copyright terms: Public domain W3C validator