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Theorem ssimaex 5555
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 4910 . . . . 5 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21imaeq2i 4949 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3 imadmres 5101 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹𝐴)
42, 3eqtr3i 2193 . . 3 (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹𝐴)
54sseq2i 3174 . 2 (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹𝐴))
6 ssrab2 3232 . . . 4 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7 ssel2 3142 . . . . . . . . 9 ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
87adantll 473 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9 fvelima 5546 . . . . . . . . . . . 12 ((Fun 𝐹𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧)
109ex 114 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
1110adantr 274 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧))
12 eleq1a 2242 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → ((𝐹𝑤) = 𝑧 → (𝐹𝑤) ∈ 𝐵))
1312anim2d 335 . . . . . . . . . . . . . . 15 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵)))
14 fveq2 5494 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1514eleq1d 2239 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ 𝐵 ↔ (𝐹𝑤) ∈ 𝐵))
1615elrab 2886 . . . . . . . . . . . . . . 15 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵))
1713, 16syl6ibr 161 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
18 simpr 109 . . . . . . . . . . . . . . 15 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧)
1918a1i 9 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧))
2017, 19jcad 305 . . . . . . . . . . . . 13 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∧ (𝐹𝑤) = 𝑧)))
2120reximdv2 2569 . . . . . . . . . . . 12 (𝑧𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2221adantl 275 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
23 funfn 5226 . . . . . . . . . . . . 13 (Fun 𝐹𝐹 Fn dom 𝐹)
24 inss2 3348 . . . . . . . . . . . . . . 15 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
256, 24sstri 3156 . . . . . . . . . . . . . 14 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹
26 fvelimab 5550 . . . . . . . . . . . . . 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))
2725, 26mpan2 423 . . . . . . . . . . . . 13 (𝐹 Fn dom 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2823, 27sylbi 120 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2928adantr 274 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
3022, 29sylibrd 168 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3111, 30syld 45 . . . . . . . . 9 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3231adantlr 474 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
338, 32mpd 13 . . . . . . 7 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
3433ex 114 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
35 fvelima 5546 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧)
3635ex 114 . . . . . . . 8 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
37 eleq1 2233 . . . . . . . . . . . 12 ((𝐹𝑤) = 𝑧 → ((𝐹𝑤) ∈ 𝐵𝑧𝐵))
3837biimpcd 158 . . . . . . . . . . 11 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) = 𝑧𝑧𝐵))
3938adantl 275 . . . . . . . . . 10 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵) → ((𝐹𝑤) = 𝑧𝑧𝐵))
4016, 39sylbi 120 . . . . . . . . 9 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝐹𝑤) = 𝑧𝑧𝐵))
4140rexlimiv 2581 . . . . . . . 8 (∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧𝑧𝐵)
4236, 41syl6 33 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4342adantr 274 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4434, 43impbid 128 . . . . 5 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
4544eqrdv 2168 . . . 4 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
46 ssimaex.1 . . . . . . 7 𝐴 ∈ V
4746inex1 4121 . . . . . 6 (𝐴 ∩ dom 𝐹) ∈ V
4847rabex 4131 . . . . 5 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∈ V
49 sseq1 3170 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
50 imaeq2 4947 . . . . . . 7 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐹𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
5150eqeq2d 2182 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐵 = (𝐹𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
5249, 51anbi12d 470 . . . . 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}))))
5348, 52spcev 2825 . . . 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)))
546, 45, 53sylancr 412 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
55 inss1 3347 . . . . . 6 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
56 sstr 3155 . . . . . 6 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥𝐴)
5755, 56mpan2 423 . . . . 5 (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥𝐴)
5857anim1i 338 . . . 4 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → (𝑥𝐴𝐵 = (𝐹𝑥)))
5958eximi 1593 . . 3 (∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
6054, 59syl 14 . 2 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
615, 60sylan2br 286 1 ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  wrex 2449  {crab 2452  Vcvv 2730  cin 3120  wss 3121  dom cdm 4609  cres 4611  cima 4612  Fun wfun 5190   Fn wfn 5191  cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by:  ssimaexg  5556
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