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Theorem rnmpt2ss 29803
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 6936 . . . 4 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32abeq2i 2873 . . 3 (𝑧 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
4 simpl 474 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
5 simpr 479 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶)
64, 5r19.29d2r 3218 . . . . 5 ((∀𝑥𝐴𝑦𝐵 𝐶𝐷 ∧ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶) → ∃𝑥𝐴𝑦𝐵 (𝐶𝐷𝑧 = 𝐶))
7 eleq1 2827 . . . . . . . 8 (𝑧 = 𝐶 → (𝑧𝐷𝐶𝐷))
87biimparc 505 . . . . . . 7 ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷)
98a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ((𝐶𝐷𝑧 = 𝐶) → 𝑧𝐷))
109rexlimivv 3174 . . . . 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 3750 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715  ran crn 5267  cmpt2 6816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-oprab 6818  df-mpt2 6819
This theorem is referenced by:  raddcn  30305  br2base  30661  sxbrsiga  30682
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