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Theorem ssimaexg 5156
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 4587 . . . . . 6 (y = A → (𝐹y) = (𝐹A))
21sseq2d 2946 . . . . 5 (y = A → (B ⊆ (𝐹y) ↔ B ⊆ (𝐹A)))
32anbi2d 440 . . . 4 (y = A → ((Fun 𝐹 B ⊆ (𝐹y)) ↔ (Fun 𝐹 B ⊆ (𝐹A))))
4 sseq2 2940 . . . . . 6 (y = A → (xyxA))
54anbi1d 441 . . . . 5 (y = A → ((xy B = (𝐹x)) ↔ (xA B = (𝐹x))))
65exbidv 1684 . . . 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 2534 . . . 4 y V
98ssimaex 5155 . . 3 ((Fun 𝐹 B ⊆ (𝐹y)) → x(xy B = (𝐹x)))
107, 9vtoclg 2586 . 2 (A 𝐶 → ((Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x))))
11103impib 1086 1 ((A 𝐶 Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 871   = wceq 1226  wex 1358   wcel 1370  wss 2890  cima 4271  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by: (None)
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