Theorem ssimaexg 5744

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 5102	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹 “ 𝑦) = (𝐹 “ 𝐴))
2	1	sseq2d 3272	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹 “ 𝑦) ↔ 𝐵 ⊆ (𝐹 “ 𝐴)))
3	2	anbi2d 464	. . . 4 ⊢ (𝑦 = 𝐴 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) ↔ (Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴))))
4		sseq2 3266	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
5	4	anbi1d 465	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
6	5	exbidv 1874	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
7	3, 6	imbi12d 234	. . 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 2818	. . . 4 ⊢ 𝑦 ∈ V
9	8	ssimaex 5743	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝑦)) → ∃𝑥(𝑥 ⊆ 𝑦 ∧ 𝐵 = (𝐹 “ 𝑥)))
10	7, 9	vtoclg 2877	. 2 ⊢ (𝐴 ∈ 𝐶 → ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))))
11	10	3impib 1228	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205   ⊆ wss 3214   “ cima 4757  Fun wfun 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  tgrest  15146