Theorem ssimaexg 6854

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

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ssimaexg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 5965	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹 “ 𝑦) = (𝐹 “ 𝐴))
2	1	sseq2d 3953	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹 “ 𝑦) ↔ 𝐵 ⊆ (𝐹 “ 𝐴)))
3	2	anbi2d 629	. . . 4 ⊢ (𝑦 = 𝐴 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) ↔ (Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴))))
4		sseq2 3947	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
5	4	anbi1d 630	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
6	5	exbidv 1924	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
7	3, 6	imbi12d 345	. . 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		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
9	8	ssimaex 6853	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) → ∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)))
10	7, 9	vtoclg 3505	. 2 ⊢ (𝐴 ∈ 𝐶 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
11	10	3impib 1115	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ⊆ wss 3887   “ cima 5592  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  tgrest  22310  cmpfi  22559  zarclsint  31822