Theorem ssimaexg 6928

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 6023	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹 “ 𝑦) = (𝐹 “ 𝐴))
2	1	sseq2d 3968	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹 “ 𝑦) ↔ 𝐵 ⊆ (𝐹 “ 𝐴)))
3	2	anbi2d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) ↔ (Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴))))
4		sseq2 3962	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
5	4	anbi1d 632	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
6	5	exbidv 1923	. . . 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 3446	. . . 4 ⊢ 𝑦 ∈ V
9	8	ssimaex 6927	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) → ∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)))
10	7, 9	vtoclg 3513	. 2 ⊢ (𝐴 ∈ 𝐶 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
11	10	3impib 1117	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3903   “ cima 5635  Fun wfun 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  tgrest  23115  cmpfi  23364  zarclsint  34049