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Theorem rnmptssrn 43298
Description: Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
rnmptssrn.b ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmptssrn.y ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)
Assertion
Ref Expression
rnmptssrn (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rnmptssrn
StepHypRef Expression
1 eqid 2737 . . . 4 (𝑦𝐶𝐷) = (𝑦𝐶𝐷)
2 rnmptssrn.y . . . 4 ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)
3 rnmptssrn.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
41, 2, 3elrnmptd 5914 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑦𝐶𝐷))
54ralrimiva 3141 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷))
6 eqid 2737 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76rnmptss 7066 . 2 (∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷) → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
85, 7syl 17 1 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  wss 3908  cmpt 5186  ran crn 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6495  df-fn 6496  df-f 6497
This theorem is referenced by:  sge0f1o  44517
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