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Theorem ssimaex 6225
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 5383 . . . . 5 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21imaeq2i 5428 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3 imadmres 5591 . . . 4 (𝐹 “ dom (𝐹𝐴)) = (𝐹𝐴)
42, 3eqtr3i 2645 . . 3 (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹𝐴)
54sseq2i 3614 . 2 (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹𝐴))
6 ssrab2 3671 . . . 4 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7 ssel2 3582 . . . . . . . . 9 ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
87adantll 749 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9 fvelima 6210 . . . . . . . . . . . 12 ((Fun 𝐹𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧)
109ex 450 . . . . . . . . . . 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 2693 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → ((𝐹𝑤) = 𝑧 → (𝐹𝑤) ∈ 𝐵))
1312anim2d 588 . . . . . . . . . . . . . . 15 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵)))
14 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1514eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ 𝐵 ↔ (𝐹𝑤) ∈ 𝐵))
1615elrab 3350 . . . . . . . . . . . . . . 15 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵))
1713, 16syl6ibr 242 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
18 simpr 477 . . . . . . . . . . . . . . 15 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧)
1918a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝐹𝑤) = 𝑧))
2017, 19jcad 555 . . . . . . . . . . . . 13 (𝑧𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∧ (𝐹𝑤) = 𝑧)))
2120reximdv2 3009 . . . . . . . . . . . 12 (𝑧𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2221adantl 482 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
23 funfn 5882 . . . . . . . . . . . . 13 (Fun 𝐹𝐹 Fn dom 𝐹)
24 inss2 3817 . . . . . . . . . . . . . . 15 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
256, 24sstri 3596 . . . . . . . . . . . . . 14 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ dom 𝐹
26 fvelimab 6215 . . . . . . . . . . . . . 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 706 . . . . . . . . . . . . 13 (𝐹 Fn dom 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2823, 27sylbi 207 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
2928adantr 481 . . . . . . . . . . 11 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
3022, 29sylibrd 249 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹𝑤) = 𝑧𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3111, 30syld 47 . . . . . . . . 9 ((Fun 𝐹𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
3231adantlr 750 . . . . . . . 8 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
338, 32mpd 15 . . . . . . 7 (((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧𝐵) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
3433ex 450 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
35 fvelima 6210 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧)
3635ex 450 . . . . . . . 8 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧))
37 eleq1 2686 . . . . . . . . . . . 12 ((𝐹𝑤) = 𝑧 → ((𝐹𝑤) ∈ 𝐵𝑧𝐵))
3837biimpcd 239 . . . . . . . . . . 11 ((𝐹𝑤) ∈ 𝐵 → ((𝐹𝑤) = 𝑧𝑧𝐵))
3938adantl 482 . . . . . . . . . 10 ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹𝑤) ∈ 𝐵) → ((𝐹𝑤) = 𝑧𝑧𝐵))
4016, 39sylbi 207 . . . . . . . . 9 (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → ((𝐹𝑤) = 𝑧𝑧𝐵))
4140rexlimiv 3021 . . . . . . . 8 (∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} (𝐹𝑤) = 𝑧𝑧𝐵)
4236, 41syl6 35 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4342adantr 481 . . . . . 6 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}) → 𝑧𝐵))
4434, 43impbid 202 . . . . 5 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧𝐵𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
4544eqrdv 2619 . . . 4 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
46 ssimaex.1 . . . . . . 7 𝐴 ∈ V
4746inex1 4764 . . . . . 6 (𝐴 ∩ dom 𝐹) ∈ V
4847rabex 4778 . . . . 5 {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ∈ V
49 sseq1 3610 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
50 imaeq2 5426 . . . . . . 7 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐹𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵}))
5150eqeq2d 2631 . . . . . 6 (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵} → (𝐵 = (𝐹𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹𝑦) ∈ 𝐵})))
5249, 51anbi12d 746 . . . . 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 3289 . . . 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 694 . . 3 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)))
55 inss1 3816 . . . . . 6 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
56 sstr 3595 . . . . . 6 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥𝐴)
5755, 56mpan2 706 . . . . 5 (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥𝐴)
5857anim1i 591 . . . 4 ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → (𝑥𝐴𝐵 = (𝐹𝑥)))
5958eximi 1759 . . 3 (∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹𝑥)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
6054, 59syl 17 . 2 ((Fun 𝐹𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
615, 60sylan2br 493 1 ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  {crab 2911  Vcvv 3189  cin 3558  wss 3559  dom cdm 5079  cres 5081  cima 5082  Fun wfun 5846   Fn wfn 5847  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860
This theorem is referenced by:  ssimaexg  6226
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