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Theorem ssimaex 6977
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 6004 . . . . 5 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21imaeq2i 6058 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3 imadmres 6234 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹𝐴)
42, 3eqtr3i 2763 . . 3 (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹𝐴)
54sseq2i 4012 . 2 (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹𝐴))
6 ssrab2 4078 . . . 4 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7 ssel2 3978 . . . . . . . . 9 ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
87adantll 713 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9 fvelima 6958 . . . . . . . . . . . 12 ((Fun 𝐹𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧)
109ex 414 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
1110adantr 482 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
12 eleq1a 2829 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → ((𝐹𝑤) = 𝑧 → (𝐹𝑤) ∈ 𝐵))
1312anim2d 613 . . . . . . . . . . . . . . 15 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵)))
14 fveq2 6892 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1514eleq1d 2819 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ 𝐵 ↔ (𝐹𝑤) ∈ 𝐵))
1615elrab 3684 . . . . . . . . . . . . . . 15 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵))
1713, 16imbitrrdi 251 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
18 simpr 486 . . . . . . . . . . . . . 14 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧)
1917, 18jca2 515 . . . . . . . . . . . . 13 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∧ (𝐹𝑤) = 𝑧)))
2019reximdv2 3165 . . . . . . . . . . . 12 (𝑧𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2120adantl 483 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
22 funfn 6579 . . . . . . . . . . . . 13 (Fun 𝐹𝐹 Fn dom 𝐹)
23 inss2 4230 . . . . . . . . . . . . . . 15 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
246, 23sstri 3992 . . . . . . . . . . . . . 14 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹
25 fvelimab 6965 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2921, 28sylibrd 259 . . . . . . . . . 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 714 . . . . . . . 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 414 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
34 fvelima 6958 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧)
3534ex 414 . . . . . . . 8 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
36 eleq1 2822 . . . . . . . . . . . 12 ((𝐹𝑤) = 𝑧 → ((𝐹𝑤) ∈ 𝐵𝑧𝐵))
3736biimpcd 248 . . . . . . . . . . 11 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) = 𝑧𝑧𝐵))
3837adantl 483 . . . . . . . . . 10 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵) → ((𝐹𝑤) = 𝑧𝑧𝐵))
3916, 38sylbi 216 . . . . . . . . 9 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝐹𝑤) = 𝑧𝑧𝐵))
4039rexlimiv 3149 . . . . . . . 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 482 . . . . . 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 2731 . . . 4 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
45 ssimaex.1 . . . . . . 7 𝐴 ∈ V
4645inex1 5318 . . . . . 6 (𝐴 ∩ dom 𝐹) ∈ V
4746rabex 5333 . . . . 5 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∈ V
48 sseq1 4008 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
49 imaeq2 6056 . . . . . . 7 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐹𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
5049eqeq2d 2744 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐵 = (𝐹𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
5148, 50anbi12d 632 . . . . 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 3597 . . . 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 588 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
54 inss1 4229 . . . . . 6 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
55 sstr 3991 . . . . . 6 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥𝐴)
5654, 55mpan2 690 . . . . 5 (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥𝐴)
5756anim1i 616 . . . 4 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → (𝑥𝐴𝐵 = (𝐹𝑥)))
5857eximi 1838 . . 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 596 1 ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  {crab 3433  Vcvv 3475  cin 3948  wss 3949  dom cdm 5677  cres 5679  cima 5680  Fun wfun 6538   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  ssimaexg  6978
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