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Theorem ssimaex 6926
Description: The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
Hypothesis
Ref Expression
ssimaex.1 𝐴 ∈ V
Assertion
Ref Expression
ssimaex ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem ssimaex
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmres 5959 . . . . 5 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21imaeq2i 6011 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3 imadmres 6186 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹𝐴)
42, 3eqtr3i 2766 . . 3 (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹𝐴)
54sseq2i 3973 . 2 (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹𝐴))
6 ssrab2 4037 . . . 4 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7 ssel2 3939 . . . . . . . . 9 ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
87adantll 712 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9 fvelima 6908 . . . . . . . . . . . 12 ((Fun 𝐹𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧)
109ex 413 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
1110adantr 481 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
12 eleq1a 2833 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → ((𝐹𝑤) = 𝑧 → (𝐹𝑤) ∈ 𝐵))
1312anim2d 612 . . . . . . . . . . . . . . 15 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵)))
14 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1514eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ 𝐵 ↔ (𝐹𝑤) ∈ 𝐵))
1615elrab 3645 . . . . . . . . . . . . . . 15 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵))
1713, 16syl6ibr 251 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
18 simpr 485 . . . . . . . . . . . . . 14 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧)
1917, 18jca2 514 . . . . . . . . . . . . 13 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∧ (𝐹𝑤) = 𝑧)))
2019reximdv2 3161 . . . . . . . . . . . 12 (𝑧𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2120adantl 482 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
22 funfn 6531 . . . . . . . . . . . . 13 (Fun 𝐹𝐹 Fn dom 𝐹)
23 inss2 4189 . . . . . . . . . . . . . . 15 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
246, 23sstri 3953 . . . . . . . . . . . . . 14 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹
25 fvelimab 6914 . . . . . . . . . . . . . 14 ((𝐹 Fn dom 𝐹 ∧ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2624, 25mpan2 689 . . . . . . . . . . . . 13 (𝐹 Fn dom 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2722, 26sylbi 216 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2827adantr 481 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2921, 28sylibrd 258 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3011, 29syld 47 . . . . . . . . 9 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3130adantlr 713 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
328, 31mpd 15 . . . . . . 7 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
3332ex 413 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
34 fvelima 6908 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧)
3534ex 413 . . . . . . . 8 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
36 eleq1 2825 . . . . . . . . . . . 12 ((𝐹𝑤) = 𝑧 → ((𝐹𝑤) ∈ 𝐵𝑧𝐵))
3736biimpcd 248 . . . . . . . . . . 11 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) = 𝑧𝑧𝐵))
3837adantl 482 . . . . . . . . . 10 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵) → ((𝐹𝑤) = 𝑧𝑧𝐵))
3916, 38sylbi 216 . . . . . . . . 9 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝐹𝑤) = 𝑧𝑧𝐵))
4039rexlimiv 3145 . . . . . . . 8 (∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧𝑧𝐵)
4135, 40syl6 35 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4241adantr 481 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4333, 42impbid 211 . . . . 5 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
4443eqrdv 2734 . . . 4 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
45 ssimaex.1 . . . . . . 7 𝐴 ∈ V
4645inex1 5274 . . . . . 6 (𝐴 ∩ dom 𝐹) ∈ V
4746rabex 5289 . . . . 5 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∈ V
48 sseq1 3969 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
49 imaeq2 6009 . . . . . . 7 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐹𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
5049eqeq2d 2747 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐵 = (𝐹𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
5148, 50anbi12d 631 . . . . 5 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) ↔ ({𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))))
5247, 51spcev 3565 . . . 4 (({𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
536, 44, 52sylancr 587 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
54 inss1 4188 . . . . . 6 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
55 sstr 3952 . . . . . 6 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥𝐴)
5654, 55mpan2 689 . . . . 5 (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥𝐴)
5756anim1i 615 . . . 4 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → (𝑥𝐴𝐵 = (𝐹𝑥)))
5857eximi 1837 . . 3 (∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
5953, 58syl 17 . 2 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
605, 59sylan2br 595 1 ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  dom cdm 5633  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  ssimaexg  6927
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