Theorem ssimaexg 6913

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 6011	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹 “ 𝑦) = (𝐹 “ 𝐴))
2	1	sseq2d 3970	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹 “ 𝑦) ↔ 𝐵 ⊆ (𝐹 “ 𝐴)))
3	2	anbi2d 630	. . . 4 ⊢ (𝑦 = 𝐴 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) ↔ (Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴))))
4		sseq2 3964	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
5	4	anbi1d 631	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
6	5	exbidv 1921	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
7	3, 6	imbi12d 344	. . 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 3442	. . . 4 ⊢ 𝑦 ∈ V
9	8	ssimaex 6912	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) → ∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)))
10	7, 9	vtoclg 3511	. 2 ⊢ (𝐴 ∈ 𝐶 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
11	10	3impib 1116	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3905   “ cima 5626  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  tgrest  23062  cmpfi  23311  zarclsint  33838