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Theorem rnmpt2ss 29307
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 6724 . . . 4 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32abeq2i 2738 . . 3 (𝑧 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
4 simpl 473 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
5 simpr 477 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
64, 5r19.29d2r 3077 . . . . 5 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶))
7 eleq1 2692 . . . . . . . 8 (𝑧 = 𝐶 → (𝑧𝐷𝐶𝐷))
87biimparc 504 . . . . . . 7 ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
98a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷))
109rexlimivv 3034 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
116, 10syl 17 . . . 4 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → 𝑧𝐷)
1211ex 450 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
133, 12syl5bi 232 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → (𝑧 ∈ ran 𝐹𝑧𝐷))
1413ssrdv 3594 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  ran crn 5080  cmpt2 6607
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-oprab 6609  df-mpt2 6610
This theorem is referenced by:  raddcn  29749  br2base  30104  sxbrsiga  30125
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