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Theorem ssimaexg 6575
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 5763 . . . . . 6 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
21sseq2d 3883 . . . . 5 (𝑦 = 𝐴 → (𝐵 ⊆ (𝐹𝑦) ↔ 𝐵 ⊆ (𝐹𝐴)))
32anbi2d 619 . . . 4 (𝑦 = 𝐴 → ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) ↔ (Fun 𝐹𝐵 ⊆ (𝐹𝐴))))
4 sseq2 3877 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
54anbi1d 620 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑦𝐵 = (𝐹𝑥)) ↔ (𝑥𝐴𝐵 = (𝐹𝑥))))
65exbidv 1880 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
73, 6imbi12d 337 . . 3 (𝑦 = 𝐴 → (((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥))) ↔ ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))))
8 vex 3412 . . . 4 𝑦 ∈ V
98ssimaex 6574 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹𝑦)) → ∃𝑥(𝑥𝑦𝐵 = (𝐹𝑥)))
107, 9vtoclg 3480 . 2 (𝐴𝐶 → ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥))))
11103impib 1096 1 ((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  wss 3823  cima 5406  Fun wfun 6179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193
This theorem is referenced by:  tgrest  21483  cmpfi  21732
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