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

Theorem ssimaexg 5483
Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
Assertion
Ref Expression
ssimaexg ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ssimaexg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 imaeq2 4877 . . . . . 6 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
21sseq2d 3127 . . . . 5 (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹𝑦) ↔ 𝐵 ⊆ (𝐹𝐴)))
32anbi2d 459 . . . 4 (𝑦 = 𝐴 → ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) ↔ (Fun 𝐹𝐵 ⊆ (𝐹𝐴))))
4 sseq2 3121 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
54anbi1d 460 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑦𝐵 = (𝐹𝑥)) ↔ (𝑥𝐴𝐵 = (𝐹𝑥))))
65exbidv 1797 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
73, 6imbi12d 233 . . 3 (𝑦 = 𝐴 → (((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥))) ↔ ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))))
8 vex 2689 . . . 4 𝑦 ∈ V
98ssimaex 5482 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)))
107, 9vtoclg 2746 . 2 (𝐴𝐶 → ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
11103impib 1179 1 ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wex 1468  wcel 1480  wss 3071  cima 4542  Fun wfun 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  tgrest  12348
  Copyright terms: Public domain W3C validator