Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2fvex Structured version   GIF version

Theorem mpt2fvex 5771
 Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpt2.1 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
mpt2fvex ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝑅𝐹𝑆) V)
Distinct variable groups:   x,A,y   x,B,y
Allowed substitution hints:   𝐶(x,y)   𝑅(x,y)   𝑆(x,y)   𝐹(x,y)   𝑉(x,y)   𝑊(x,y)   𝑋(x,y)

Proof of Theorem mpt2fvex
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-ov 5458 . 2 (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2 elex 2560 . . . . . . . . 9 (𝐶 𝑉𝐶 V)
32alimi 1341 . . . . . . . 8 (y 𝐶 𝑉y 𝐶 V)
4 vex 2554 . . . . . . . . 9 z V
5 2ndexg 5737 . . . . . . . . 9 (z V → (2ndz) V)
6 nfcv 2175 . . . . . . . . . 10 y(2ndz)
7 nfcsb1v 2876 . . . . . . . . . . 11 y(2ndz) / y𝐶
87nfel1 2185 . . . . . . . . . 10 y(2ndz) / y𝐶 V
9 csbeq1a 2854 . . . . . . . . . . 11 (y = (2ndz) → 𝐶 = (2ndz) / y𝐶)
109eleq1d 2103 . . . . . . . . . 10 (y = (2ndz) → (𝐶 V ↔ (2ndz) / y𝐶 V))
116, 8, 10spcgf 2629 . . . . . . . . 9 ((2ndz) V → (y 𝐶 V → (2ndz) / y𝐶 V))
124, 5, 11mp2b 8 . . . . . . . 8 (y 𝐶 V → (2ndz) / y𝐶 V)
133, 12syl 14 . . . . . . 7 (y 𝐶 𝑉(2ndz) / y𝐶 V)
1413alimi 1341 . . . . . 6 (xy 𝐶 𝑉x(2ndz) / y𝐶 V)
15 1stexg 5736 . . . . . . 7 (z V → (1stz) V)
16 nfcv 2175 . . . . . . . 8 x(1stz)
17 nfcsb1v 2876 . . . . . . . . 9 x(1stz) / x(2ndz) / y𝐶
1817nfel1 2185 . . . . . . . 8 x(1stz) / x(2ndz) / y𝐶 V
19 csbeq1a 2854 . . . . . . . . 9 (x = (1stz) → (2ndz) / y𝐶 = (1stz) / x(2ndz) / y𝐶)
2019eleq1d 2103 . . . . . . . 8 (x = (1stz) → ((2ndz) / y𝐶 V ↔ (1stz) / x(2ndz) / y𝐶 V))
2116, 18, 20spcgf 2629 . . . . . . 7 ((1stz) V → (x(2ndz) / y𝐶 V → (1stz) / x(2ndz) / y𝐶 V))
224, 15, 21mp2b 8 . . . . . 6 (x(2ndz) / y𝐶 V → (1stz) / x(2ndz) / y𝐶 V)
2314, 22syl 14 . . . . 5 (xy 𝐶 𝑉(1stz) / x(2ndz) / y𝐶 V)
2423alrimiv 1751 . . . 4 (xy 𝐶 𝑉z(1stz) / x(2ndz) / y𝐶 V)
25243ad2ant1 924 . . 3 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → z(1stz) / x(2ndz) / y𝐶 V)
26 opexg 3955 . . . 4 ((𝑅 𝑊 𝑆 𝑋) → ⟨𝑅, 𝑆 V)
27263adant1 921 . . 3 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → ⟨𝑅, 𝑆 V)
28 fmpt2.1 . . . . 5 𝐹 = (x A, y B𝐶)
29 mpt2mptsx 5765 . . . . 5 (x A, y B𝐶) = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
3028, 29eqtri 2057 . . . 4 𝐹 = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
3130mptfvex 5199 . . 3 ((z(1stz) / x(2ndz) / y𝐶 V 𝑅, 𝑆 V) → (𝐹‘⟨𝑅, 𝑆⟩) V)
3225, 27, 31syl2anc 391 . 2 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) V)
331, 32syl5eqel 2121 1 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝑅𝐹𝑆) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 884  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⦋csb 2846  {csn 3367  ⟨cop 3370  ∪ ciun 3648   ↦ cmpt 3809   × cxp 4286  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1st c1st 5707  2nd c2nd 5708 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-13 1401  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  ax-un 4136 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-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-iun 3650  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-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710 This theorem is referenced by:  mpt2fvexi  5774  oaexg  5967  omexg  5970  oeiexg  5972
 Copyright terms: Public domain W3C validator