Theorem ssimaexg 6264

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 5462	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹 “ 𝑦) = (𝐹 “ 𝐴))
2	1	sseq2d 3633	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹 “ 𝑦) ↔ 𝐵 ⊆ (𝐹 “ 𝐴)))
3	2	anbi2d 740	. . . 4 ⊢ (𝑦 = 𝐴 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) ↔ (Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴))))
4		sseq2 3627	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
5	4	anbi1d 741	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
6	5	exbidv 1850	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
7	3, 6	imbi12d 334	. . 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 3203	. . . 4 ⊢ 𝑦 ∈ V
9	8	ssimaex 6263	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) → ∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)))
10	7, 9	vtoclg 3266	. 2 ⊢ (𝐴 ∈ 𝐶 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
11	10	3impib 1262	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ⊆ wss 3574   “ cima 5117  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  tgrest  20963  cmpfi  21211