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Theorem elimampt 6061
Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Hypotheses
Ref Expression
elimampt.f 𝐹 = (𝑥𝐴𝐵)
elimampt.c (𝜑𝐶𝑊)
elimampt.d (𝜑𝐷𝐴)
Assertion
Ref Expression
elimampt (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem elimampt
StepHypRef Expression
1 df-ima 5698 . . 3 (𝐹𝐷) = ran (𝐹𝐷)
21eleq2i 2833 . 2 (𝐶 ∈ (𝐹𝐷) ↔ 𝐶 ∈ ran (𝐹𝐷))
3 elimampt.d . . . 4 (𝜑𝐷𝐴)
4 elimampt.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54reseq1i 5993 . . . . . . 7 (𝐹𝐷) = ((𝑥𝐴𝐵) ↾ 𝐷)
6 resmpt 6055 . . . . . . 7 (𝐷𝐴 → ((𝑥𝐴𝐵) ↾ 𝐷) = (𝑥𝐷𝐵))
75, 6eqtrid 2789 . . . . . 6 (𝐷𝐴 → (𝐹𝐷) = (𝑥𝐷𝐵))
87rneqd 5949 . . . . 5 (𝐷𝐴 → ran (𝐹𝐷) = ran (𝑥𝐷𝐵))
98eleq2d 2827 . . . 4 (𝐷𝐴 → (𝐶 ∈ ran (𝐹𝐷) ↔ 𝐶 ∈ ran (𝑥𝐷𝐵)))
103, 9syl 17 . . 3 (𝜑 → (𝐶 ∈ ran (𝐹𝐷) ↔ 𝐶 ∈ ran (𝑥𝐷𝐵)))
11 elimampt.c . . . 4 (𝜑𝐶𝑊)
12 eqid 2737 . . . . 5 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1312elrnmpt 5969 . . . 4 (𝐶𝑊 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1411, 13syl 17 . . 3 (𝜑 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1510, 14bitrd 279 . 2 (𝜑 → (𝐶 ∈ ran (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
162, 15bitrid 283 1 (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cmpt 5225  ran crn 5686  cres 5687  cima 5688
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  reprpmtf1o  34641  ellcsrspsn  35646
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