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Theorem ssimaex 6975
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 6002 . . . . 5 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21imaeq2i 6056 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3 imadmres 6232 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹𝐴)
42, 3eqtr3i 2760 . . 3 (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹𝐴)
54sseq2i 4010 . 2 (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹𝐴))
6 ssrab2 4076 . . . 4 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7 ssel2 3976 . . . . . . . . 9 ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
87adantll 710 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9 fvelima 6956 . . . . . . . . . . . 12 ((Fun 𝐹𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧)
109ex 411 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
1110adantr 479 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
12 eleq1a 2826 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → ((𝐹𝑤) = 𝑧 → (𝐹𝑤) ∈ 𝐵))
1312anim2d 610 . . . . . . . . . . . . . . 15 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵)))
14 fveq2 6890 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1514eleq1d 2816 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ 𝐵 ↔ (𝐹𝑤) ∈ 𝐵))
1615elrab 3682 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧)
1917, 18jca2 512 . . . . . . . . . . . . 13 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∧ (𝐹𝑤) = 𝑧)))
2019reximdv2 3162 . . . . . . . . . . . 12 (𝑧𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2120adantl 480 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
22 funfn 6577 . . . . . . . . . . . . 13 (Fun 𝐹𝐹 Fn dom 𝐹)
23 inss2 4228 . . . . . . . . . . . . . . 15 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
246, 23sstri 3990 . . . . . . . . . . . . . 14 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹
25 fvelimab 6963 . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . 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 479 . . . . . . . . . . 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 711 . . . . . . . 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 411 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
34 fvelima 6956 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧)
3534ex 411 . . . . . . . 8 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
36 eleq1 2819 . . . . . . . . . . . 12 ((𝐹𝑤) = 𝑧 → ((𝐹𝑤) ∈ 𝐵𝑧𝐵))
3736biimpcd 248 . . . . . . . . . . 11 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) = 𝑧𝑧𝐵))
3837adantl 480 . . . . . . . . . 10 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵) → ((𝐹𝑤) = 𝑧𝑧𝐵))
3916, 38sylbi 216 . . . . . . . . 9 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝐹𝑤) = 𝑧𝑧𝐵))
4039rexlimiv 3146 . . . . . . . 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 479 . . . . . 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 2728 . . . 4 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
45 ssimaex.1 . . . . . . 7 𝐴 ∈ V
4645inex1 5316 . . . . . 6 (𝐴 ∩ dom 𝐹) ∈ V
4746rabex 5331 . . . . 5 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∈ V
48 sseq1 4006 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
49 imaeq2 6054 . . . . . . 7 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐹𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
5049eqeq2d 2741 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐵 = (𝐹𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
5148, 50anbi12d 629 . . . . 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 3595 . . . 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 585 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
54 inss1 4227 . . . . . 6 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
55 sstr 3989 . . . . . 6 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥𝐴)
5654, 55mpan2 687 . . . . 5 (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥𝐴)
5756anim1i 613 . . . 4 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → (𝑥𝐴𝐵 = (𝐹𝑥)))
5857eximi 1835 . . 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 593 1 ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wrex 3068  {crab 3430  Vcvv 3472  cin 3946  wss 3947  dom cdm 5675  cres 5677  cima 5678  Fun wfun 6536   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  ssimaexg  6976
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