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Theorem ssimaexg 5178
Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
Assertion
Ref Expression
ssimaexg ((A 𝐶 Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (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 4607 . . . . . 6 (y = A → (𝐹y) = (𝐹A))
21sseq2d 2967 . . . . 5 (y = A → (B ⊆ (𝐹y) ↔ B ⊆ (𝐹A)))
32anbi2d 437 . . . 4 (y = A → ((Fun 𝐹 B ⊆ (𝐹y)) ↔ (Fun 𝐹 B ⊆ (𝐹A))))
4 sseq2 2961 . . . . . 6 (y = A → (xyxA))
54anbi1d 438 . . . . 5 (y = A → ((xy B = (𝐹x)) ↔ (xA B = (𝐹x))))
65exbidv 1703 . . . 4 (y = A → (x(xy B = (𝐹x)) ↔ x(xA B = (𝐹x))))
73, 6imbi12d 223 . . 3 (y = A → (((Fun 𝐹 B ⊆ (𝐹y)) → x(xy B = (𝐹x))) ↔ ((Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))))
8 vex 2554 . . . 4 y V
98ssimaex 5177 . . 3 ((Fun 𝐹 B ⊆ (𝐹y)) → x(xy B = (𝐹x)))
107, 9vtoclg 2607 . 2 (A 𝐶 → ((Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x))))
11103impib 1101 1 ((A 𝐶 Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wex 1378   wcel 1390  wss 2911  cima 4291  Fun wfun 4839
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-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-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-fv 4853
This theorem is referenced by: (None)
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