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

Theorem ssimaexg 5599
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 4984 . . . . . 6 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
21sseq2d 3200 . . . . 5 (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹𝑦) ↔ 𝐵 ⊆ (𝐹𝐴)))
32anbi2d 464 . . . 4 (𝑦 = 𝐴 → ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) ↔ (Fun 𝐹𝐵 ⊆ (𝐹𝐴))))
4 sseq2 3194 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
54anbi1d 465 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑦𝐵 = (𝐹𝑥)) ↔ (𝑥𝐴𝐵 = (𝐹𝑥))))
65exbidv 1836 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
73, 6imbi12d 234 . . 3 (𝑦 = 𝐴 → (((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥))) ↔ ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))))
8 vex 2755 . . . 4 𝑦 ∈ V
98ssimaex 5598 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)))
107, 9vtoclg 2812 . 2 (𝐴𝐶 → ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
11103impib 1203 1 ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wex 1503  wcel 2160  wss 3144  cima 4647  Fun wfun 5229
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  tgrest  14146
  Copyright terms: Public domain W3C validator