Theorem ssimaexg 3775

Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Assertion

Ref	Expression
ssimaexg	⊢ ((A ∈ C ⋀ Fun F ⋀ B ⊆ (F “ A)) → ∃x(x ⊆ A ⋀ B = (F “ x)))

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem ssimaexg

Step	Hyp	Ref	Expression
1		imaeq2 3408	. . . . . 6 ⊢ (y = A → (F “ y) = (F “ A))
2	1	sseq2d 2092	. . . . 5 ⊢ (y = A → (B ⊆ (F “ y) ↔ B ⊆ (F “ A)))
3	2	anbi2d 618	. . . 4 ⊢ (y = A → ((Fun F ⋀ B ⊆ (F “ y)) ↔ (Fun F ⋀ B ⊆ (F “ A))))
4		sseq2 2086	. . . . . 6 ⊢ (y = A → (x ⊆ y ↔ x ⊆ A))
5	4	anbi1d 619	. . . . 5 ⊢ (y = A → ((x ⊆ y ⋀ B = (F “ x)) ↔ (x ⊆ A ⋀ B = (F “ x))))
6	5	exbidv 1281	. . . 4 ⊢ (y = A → (∃x(x ⊆ y ⋀ B = (F “ x)) ↔ ∃x(x ⊆ A ⋀ B = (F “ x))))
7	3, 6	imbi12d 628	. . 3 ⊢ (y = A → (((Fun F ⋀ B ⊆ (F “ y)) → ∃x(x ⊆ y ⋀ B = (F “ x))) ↔ ((Fun F ⋀ B ⊆ (F “ A)) → ∃x(x ⊆ A ⋀ B = (F “ x)))))
8		visset 1816	. . . 4 ⊢ y ∈ V
9	8	ssimaex 3774	. . 3 ⊢ ((Fun F ⋀ B ⊆ (F “ y)) → ∃x(x ⊆ y ⋀ B = (F “ x)))
10	7, 9	vtoclg 1850	. 2 ⊢ (A ∈ C → ((Fun F ⋀ B ⊆ (F “ A)) → ∃x(x ⊆ A ⋀ B = (F “ x))))
11	10	3impib 833	1 ⊢ ((A ∈ C ⋀ Fun F ⋀ B ⊆ (F “ A)) → ∃x(x ⊆ A ⋀ B = (F “ x)))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∃wex 982   ⊆ wss 2050   “ cima 3179  Fun wfun 3182

This theorem is referenced by:  subtop 7643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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