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Theorem ssimaexg 6734
 Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
Assertion
Ref Expression
ssimaexg ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
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.
StepHypRef Expression
1 imaeq2 5896 . . . . . 6 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
21sseq2d 3949 . . . . 5 (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹𝑦) ↔ 𝐵 ⊆ (𝐹𝐴)))
32anbi2d 631 . . . 4 (𝑦 = 𝐴 → ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) ↔ (Fun 𝐹𝐵 ⊆ (𝐹𝐴))))
4 sseq2 3943 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
54anbi1d 632 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑦𝐵 = (𝐹𝑥)) ↔ (𝑥𝐴𝐵 = (𝐹𝑥))))
65exbidv 1922 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
73, 6imbi12d 348 . . 3 (𝑦 = 𝐴 → (((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥))) ↔ ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))))
8 vex 3445 . . . 4 𝑦 ∈ V
98ssimaex 6733 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)))
107, 9vtoclg 3516 . 2 (𝐴𝐶 → ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
11103impib 1113 1 ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ⊆ wss 3883   “ cima 5526  Fun wfun 6326 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-fv 6340 This theorem is referenced by:  tgrest  21805  cmpfi  22054  zarclsint  31291
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