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Theorem ssimaexg 6174
 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 5381 . . . . . 6 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
21sseq2d 3596 . . . . 5 (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹𝑦) ↔ 𝐵 ⊆ (𝐹𝐴)))
32anbi2d 736 . . . 4 (𝑦 = 𝐴 → ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) ↔ (Fun 𝐹𝐵 ⊆ (𝐹𝐴))))
4 sseq2 3590 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
54anbi1d 737 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑦𝐵 = (𝐹𝑥)) ↔ (𝑥𝐴𝐵 = (𝐹𝑥))))
65exbidv 1837 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
73, 6imbi12d 333 . . 3 (𝑦 = 𝐴 → (((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥))) ↔ ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))))
8 vex 3176 . . . 4 𝑦 ∈ V
98ssimaex 6173 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)))
107, 9vtoclg 3239 . 2 (𝐴𝐶 → ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
11103impib 1254 1 ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540   “ cima 5041  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  tgrest  20773  cmpfi  21021
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