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Theorem rnmposs 32403
Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
Hypothesis
Ref Expression
rnmposs.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
rnmposs (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem rnmposs
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rnmposs.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpo 7537 . . . 4 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32eqabri 2871 . . 3 (𝑧 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
4 2r19.29 3133 . . . . 5 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶))
5 eleq1 2815 . . . . . . . 8 (𝑧 = 𝐶 → (𝑧𝐷𝐶𝐷))
65biimparc 479 . . . . . . 7 ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
76a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷))
87rexlimivv 3193 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
94, 8syl 17 . . . 4 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → 𝑧𝐷)
109ex 412 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
113, 10biimtrid 241 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (𝑧 ∈ ran 𝐹𝑧𝐷))
1211ssrdv 3983 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  ran crn 5670  cmpo 7406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  fedgmul  33233  raddcn  33438  br2base  33797  sxbrsiga  33818
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