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Theorem rnmpt2ss 29601
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
rnmpt2ss.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
rnmpt2ss (∀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 rnmpt2ss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rnmpt2ss.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpt2 6812 . . . 4 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32abeq2i 2764 . . 3 (𝑧 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
4 simpl 472 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
5 simpr 476 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
64, 5r19.29d2r 3109 . . . . 5 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶))
7 eleq1 2718 . . . . . . . 8 (𝑧 = 𝐶 → (𝑧𝐷𝐶𝐷))
87biimparc 503 . . . . . . 7 ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
98a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷))
109rexlimivv 3065 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
116, 10syl 17 . . . 4 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → 𝑧𝐷)
1211ex 449 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
133, 12syl5bi 232 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (𝑧 ∈ ran 𝐹𝑧𝐷))
1413ssrdv 3642 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  ran crn 5144  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  raddcn  30103  br2base  30459  sxbrsiga  30480
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