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Theorem rnmpt2ss 28856
 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 6668 . . . 4 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32abeq2i 2722 . . 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 3061 . . . . 5 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶))
7 eleq1 2676 . . . . . . . 8 (𝑧 = 𝐶 → (𝑧𝐷𝐶𝐷))
87biimparc 503 . . . . . . 7 ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
98a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷))
109rexlimivv 3018 . . . . 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 231 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (𝑧 ∈ ran 𝐹𝑧𝐷))
1413ssrdv 3574 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ran crn 5039   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  raddcn  29303  br2base  29658  sxbrsiga  29679
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